The works of Edmund Gunter : containing the description and use of the sector, cross-staff, bow, quadrant, and other instruments : with a canon of artificial sines and tangents to a radius of 10.00000 parts, and the logarithms from an unite to 10000 : the uses whereof are illustrated in the practice of arithmetick, geometry, astronomy, navigation, dialling, and fortifcation : and some questions in navigation added by Mr. Henry Bond ... : to which is added, the description and use of another sector and quadrant, both of them invented by Mr. Sam. Foster ..., furnished with more lines, and differing from those of Mr. Gunters both in form and manner of working. --

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BSN DEH Fut

PURCHASED FOR THE
UNIVERSITY OF TORONTO LIBRARY
FROM THE

HUMANITIES RESEARCH COUNCIL
SPEGIAIGRAN I)

FOR

Galileo

STILLMAN DRAKE
——<———————— dl
|

Digitized by the Internet Archive
in 2024 with funding from
University of Toronto

https://archive.org/details/worksofedmundgunOOgunt_0O

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EDMUND GUNTER:

Containing the Defcription and Ufe of the

Sector, Crofs-ftaff. Bow, Quadran t

And other Inftrumenss.

With a Canon of Artificial Sines and Tangents to a
Radiusof 10,08000 parts, and the Logarithms
from an Unite to yoooo:

| The Ufes whereof are illuftrated in the Pra@ice of

Avithmetick, 5 irene, 23 Dialling, and
Geometry, Navigation, § € Fortification.
And fome Queftions in Navipation added by Mr, Hary Bond, Teacher of .
3 ~ Mathematicks in Ratcliff, near London.

=

To which js added,

The Defcription and Ufe of another Se&tor and Quadrant,
both of them invented by Mr. Sam. Fofter, Late Profeffor of Aftronomy
in Grefham Colledge, London, furnithed with more Lines, and differing
from thofe of Mr. Gunters both in form and manner of Working.

_~ oe

| Lhe Fifth Enition,
Diligently Correéted, and divers neceflary Things and-Matters ( pertinent
‘thereunto )added, throughout the whole work, not before Printed.

By Wiliam Leybourn, Philomath.
ONDON,

: L
Printed by .4,c. for Francis Felesteld at the Marigold in.
St. Pauls Church-yard. iD CLXXTITI,

‘vei yor inh
slink wan

TO THE
RIGHT HONOURABLE.

J O

Farl of Bridgwater, Vifcount Brackley,
Baron Elle{more : One of the Lords
of His Majefties moft Honourable
Privie Council, and Lord Lieutenant.
of the County of Buckingham.

S

Pe 4

My Lord,

SEDC Hele Works of the Learned Gunter, do.
Var naturally and of right, addrefi them-
A mM) felves to your Eonours Patronage: ha-
@ ving been originally Infignalized with»
zo the Titles of your Renowned Anceftors,
: oa «under whom (near Fifty. years fince )-
they recetyed their firft Life: So that they feemto be Ins.
tatled on your Illuftrions Family: Elpecially, confidering
the whole World owns your Lord/hip, no lefs Heir to your.
Ancefters refplendent Demeans , then of their love to No-.
ble Arts, amongst which thofe of the. Atathematichs, ss.
noneof the meane/? you are Mafter of.

My;

at eS ee AMT | WS Se eee “Arias
5 se Bxees sibaeg Pe moet ean SR sa Hea of eee ‘
“a

"The Epiftle Dedicatory. —
My Lord, The World taking example by fo good aCom-
—— mendum has encouraged this Work to this Fifth Edition.
= Bue it having met with the ill ufage in former Impre[st-

i. ons to have contratted feveral Typographical Sphalmata,
; which did fomewbat disfigure its beauty, I thought it
requifite to beftow fome pains in their removal, as alfoto
; furnifh it with the attendance of fome compleat Traéts of

Mr. Samuel Fofter our Learned Authors Succeffor in bis
Afironomical Profe(Sion in Grefham Colledge : What
otber Additions I have made to the Work, in Jeveral places,
He which are not only pertinent, but neceffary, the Reader
id is acquainted With in the Preface. My Lord, This Work
being arrived to this fate of PerfeCtion, pleads for a bolder
acces to your Elonotirs hands; and makes it humbly con-
fident to find your Honour no les favourable to it, now

r grown up, than your Predeceffors have been to its in-
fancy. My Lord, I have derived, likewife, hence fome

a [hare of that humble confidence, that your Efonour will
pardon this prefumption, of my fabfcribing my félf,

| My Lord,

j Your Lordfhips moft obliged,

. and obfequious Servant,

- eg as, ce peek
4 cae Viliam Leybourn,

CPR NEIL de etek oke SERIE OL So ROR. yn Mie Oe MRE Ok ee
Sete ete pat me are PEP ST ARN ve at 4 Sea 2

WILLIAM LEYBOURN

| TO THE
BM aced Am far from the vanity of defiring to

bed
4 Face have it thought,that I prefix my Name
GE ty") asa Bulh or Garland to invite any to
P39 (eaerg to the Purchafing of this Book; The.
Puss Learned Authors Authority is more
than J or any other ean fay for it, and the number
of Imprefsions that have been fo welcomed by the
Publick is afufhicient Te/timony of its good acceptance
_ in the World, for indeed, of all the Mathematical Books
yet extant, [know not one more full of Variety of
matter, nor more Praéfical chan this js. |
All that I defignin this Preface is an Apology for
my felf, to ask pardon of the more knowing Mathe-
matician, for my confidence in prefuming to fhelter
_ any of my mean and Weak Performances under the Ca-
nopy of fo profound a Ma/ter of Mathematical Learning
as this our Author was. But to fuch as fhall be
offended therewith (as, Ihope, none juftly can) let
me fay thus much for my felf : |
a) tl | é

A

Bye
i

blamed.

To the Reader.
1. Fam not the firft that ( with good fuccefS ) have
attempted the like. 7
2. In what I have done inthis Work, I have not
diminifhed or expunged one Syllable of the learned
Authors, but retained his own Method, and the feveral
Examples throughout the Book I have carefully exami-

ned, and where { found any Typographical Error, 1

made bold to correétit, for which,| prelume,{ deferve
rather Thanks than Blame.

3. That whatfoever herein J have attempted to
infert, is nothing but what is abfolutely pertinent to
our Authors Works, and renders his Inftruments to
young Tyroes in thefe Sciences more ufeful than they
could otherwife imagine. |

4. In what part of this Book foever I have aided
any ching, I have done the Author chis right, for inthe
Coneents before the Book, relating to the Page
wherein any Inferfion of mine is, I have before it
placed the figure of a band pointing thus £3. : So that
if I have done any thing, misbecoming an Artiff, the
Author may not be charged with it, but my felf juftly

And although, there are here and there fome hints
of things in feveral places of the Book of mine in-
ferted, yet the principal are thele, viz,

‘4. Inthe SECTOR, where (after our Author -

hath

To the Reader,
hath treated of Projeéting of the Sphere in Plano u pon
all the principal Spherical (Circles ) I haye added one
other Projeétion upon an Oblique Circle, wherein Cif
I deceive not my {elf ) I have given more light to Pro-
jeébion in Plano,chan is yet extant in our Mother Tongue:
for out of this Oblique Projection may be demonftra-
ted the whole rt of Dialling, and infome meafure ié
is there effected.

2. Inthe CROSS-STA FF (afterour Author
hath treated of the Menfuration of Plain Regular Su-
perficies )1 have inferted the Menfuration of {uch as
are not Uniform, as alfo of Multangulars, Regular
Poligons, &c. And (after his Menfuration of Regular
fquared Solids ) 1 have added the Menfuration of
Pri/mes, Pyramids, and Cones, both whole and diffeéted.
And with thele and fuch like neceffary matters, Ihave
in feveral other places {upplied a Vacancy.

- To the fecond Appendix, which is the ufe of a
Quadrant, of Mr. Samuel Fofters Invention, Printed
with the former Edition of thefe our Authors Works,
I have altered nothing, but have added the (on/truéti-
on of the fame Quadrant formerly wholly omitted.
And in his Alteration of the SEC T OR, I have cor-
refted {ome Overfights, and miftakes, which werein
_the former Edition ( that being Printed by aCopy —
lels Correct ) by the help of Mr. Fo/fers own Mane

Ca 2) foripe,

ee

T 0 the Reader:

jevipt, which I was accommodated with from the
worthy Dr. Fobn Twifden, a moft induftrious Mathe-
matician, and a worthy honourer of the Learned
Mr. Foffer, to whom (not only my felf, buc) the
whole World in general is engaged for his care and
pains in the Publication of divers of Mr. Foflers
Works with feveral of bis own both in Latane and Engs’
lifh ina Book Entituled Mifeellanies, or Mathematical

Lucubrations of Mt. Samuel Fofter.
Having thus far declared my (elf, and endeavoured
to take off fuch alperfions as might poffibly have been.
thrown upon me, Give me leave ( for the Dead
cannot plead for themfelves ) to take notice of fome
Plagiaries and Purlciners of other mens Labours and
Ingenuities, who out of Lucre to themfelyes, and Emu-
lation to.others of better parts, have lately thrown in-
co the World ( tothe grand abufe thereof ) feveral
trivial Traétates, extracted ( or rather tran{cribed )
both ftom our Author, and alfo from the Works and
Manufcripts of the fore-mentioned Mr. Foffer, our
Authors Succeflor in the Affronomical Profe/%ion in Gre-
{ham Colledge,London, Publifhing them to the World
in theirown names, without taking the leaft notice.
of the learned Authors, whence they originally filtcht
thofe ornaments wherewith they pride themfelves in
their feveral Pamphlets, not fo much as mentioning
| their

To the Reader,
_ their names with any duerefpect. Ineed not tell thee
who they be, Their own Impertinencies having
made them notorious enough, for fome of them
( rather than they will wane applaufe ) become their
own Encomiafters, founding their own Trumpets be-
foretheir Books, both in Englifh, Greek, and Latine.
But leaving thele to the juft cenfure of all that hall
_take due notice of them, give me leave to commend
thee to the perufal of thefe Works of our Judicious
Authors, in the Ufe and Praéice whereof (as in all.
otherthy honeft Attempts and Endeavours ) Lwifhthee
good fucce/s, and fo for this time bid thee |

April 18. ° | Farewel,.
1673. f |

Pavigation 3 The Principles thereof, and the 3 evel eC

Profefled and Taught by Wiliam Leybourn.

3 ‘ In Whole Numbers, and Fra@ions, .
Arithinetick, ; In Decimals, and by Logarithms, }
Inftrumentally, by Decimal Scales, Napiers Bones: and to
extract the Square and Cube Roots by Infpe&ion.

hy Praétice :
Geometry : cites ial pha ’

with the ; |
, Demonftration. i
The Defcription of the Circles of the Sphere.
3 Celeftial, and
Atirononry: The Ufe of the Globes, 9 Terreftrial. ie
aa To projeé the Sphere iz Plano upon any Circle, Gunna:
And upon thefe Foundations the following Superftructures.

Heights, Trees, Towers,¢g°c.
(Longimetriz, or the ) Depths, ‘ Mines, Wells, De-
Menfuration of aa Go {cents, (7.
~The Ue of | ’ Diftances,_) © (Churches, Towers, ee.

Anfrumentg, 4 Menfurationof SP aver Ste) Or any ebner Superficies,
in. the Tiling,@e.J) |
| : . ig? Timber, growing or fquared,
Pradice of es ieee 7 iy Stone, regular or irregular,
Pilate SE Mid Cask, commonly called Gauging.
UGeodefia,or the Meafuring of Land divers ways, and by feveral
Inftruments ; to draw the Plot of a whole Mannor or Lord=
fhip ; to caft up the Content thereof ; and to beautifie the
fame with all neceffary Ornaments thereunto belonging,
Or, the Menfuration of ‘Triangles, both cue
Wrigonometrta ; Geometry.
Nag Aftronomy,
The Application thereof, in the folution JB Geography.
of Problems in Navigation.
Fortification,
| Dialling,¢yc.
The Plain SeaChare

, | Board,
Geometrical | prncmerris, or ase

manner of Sailing by The Arch of a great Circle,

Sines.
Bee Arithmetically, by the Tables of ; Tangents,
Wozologiographra, Logarithms.
Or |

Geometrically, by 3 caret fe

Inffrumentally, by the Se@or, Quadrants,Scales,and other In-
ftruments, accommodated with Lines for that purpofe,

You may hear of him at Mr.Hayes’s at the Croft-daggers in Moor-fields.

Dialling :

ray va!

aS

Arts and Sciences MATHEMAT! CA Le

CONTENTS

‘The F irft Book of the Se&or.

CHAP, I.

a defcription, making, and general ufe of the Sector Page 1
Chap. 2, The ufe of the Scale of Lines , 17

To fet down a Line'refembling any given parts, or frattion of parts
Ibi¢,
To increafe, or diminifh a Line in agiven proportion 18
To divide a Line into parts given Ibid.
To find 4 proportion between two or more right Lines given 19
Two Lines being given, to find a third in continsal proportion 20
Three Lines being giveny to find a fourth in difcontinnal proportion 21
To divide a Line in (uch fort, as another Line 4s before divided 22
Two Numbers being given, to find a third in continsal proportion 23
Three Numbers being given, to find a fourth in difcontinual proportion 24
Chap. 3. The ufeof the Lines of Superficies 1. To find a proportion between
two or more like Superficies 26
To augment, or diminilh a Superficies ina given proportion Ibid,
To add one like Superficies to another. To fubtratt one like Superficies from
another : 27
To find a mzan proportional between two Lines given 28
To make a Square equaltoaS uperficies given Ibid.
To find a proportion betwees Superficies, though they be wnlike one to another
29

Toe

Sa J
3 ee ee

-

| 3§
Chap. 4. The #fe of the Lines of Solids. To find a proportion between two, or

st aa
To add one like Solid to another, To [ubtratt one like Solid from another

sh i FP a MER Me eR EN A) Ha ee sre ne EN, Po RTS TOR? "
fie (oa /
of r = +
~

The C ongents?

* Fomake a Superficies like to one Superficiess and equal to another Ibid.

To find a mean Proportional between two Numbers given

: 30
To find the {quare root of a Numbers the root being givens to find she (q4are

Number of that root : Aah 3t
Three Nuwbers being given, to find a fourth in a duplicated proportion

32
8S To defcribe a Parabola by help of the Line of Lines and Superficies

more like Solids | 36

To angment, or dimini(h a Solid, in a given proportion

‘Ibid.

To find two mean proportional Lines between two extreme Lines given
“ . 3§
To find the like Number between two Numbers 39
To find the Cubick root of a Number. ea
Three Numbers being given, to find the fourth in a triplicated proportion
| 4t

The Contents of the fecond Book of the Sean

came © i the nature of Sines, Chords, Tangents, and Secants

| Page 43
Chap.2. 7 he general ufe of Sines and tangents . a6
The Radius being known, to find the right Sine of any Ark or Angle Abid.

‘The right Sine of any Ark being given, to find the Radivs

47

The Radius of a (Circle, or the right Sine of any Ark given, anda fright
Line refembling a Sine, to find the qaantity of ‘that, unknown Sine

: Ibi¢,

The Radius, or any right Sine given, to find the verfed Sine of any Ark
| 8

ak ‘ 4
‘Having the Diameter, or Semidiarseter of aCircle, to find the Chords of

every Ark | [bic
Having two right Lines refemsbling the Chords, and ver{ed Sine, to find the
Dismeter and Radius —~ : 5°
Th: Chord of any Ark being given, to find the Diameter and Radius 5
: Having

| The Cotitents: ee
Having the Dianseters of an Ellipfis, to deferibe the fame upon & Plane

52

- To open the Setter to any Angle, or the Seétor being opened, to find the guan-
tity of the Angle 54
To find the quantity of any Angle given abies
Upon aright Line, and a point given in it, to make ats Angle equal to any

Angle given | 56
To divide the Circumference of a Circle into parts given Ibid.
T0 divide a right Line by extreme and mean proportion sry $F
Chap. 3. Of the Projection of the Sphere in plano 58
With a Nottarnal to (bew the hour ft the night 64
Ee The ufe of the Horizontal Projeétion in Dialling 7X
53> To Projekt the Sphere npon Oblique Circles ine e?
Chap.4. Of the refolution of right Line Priangles 76

Chap. 5. Of the refolution of Spherical Triangles, iz 28 cafes 85
Chap. 6. Of the fi of the Meridian Line. To divide a Sea-chart after

Mercacors Projection, with a Table to that parpofe 99
To find how many Leagues anfwer to one degree of Longitude in every feveral

Latitude 115
To find how many Leagues an[wer to one degree of Latitude, in every feveral
Ram 113
By one Latitude, Ramb, and diftance, to find the difference of Latitudes
| a5)
By the Rumb and both Latitudes, to find the dif-ance mpon the Rumb
116

By the diftance and both Latitudes, to find the Ramb 120
By the Longitude and Latitude of two places, to find the Rumb 12%

By the Runsh and both Latitudes, tofind the difference of Longitude, with
Several T ables to this purpofe, which may aifo ferve for drawing off the

Reumbs upon any Chart er Globe 122
By the difference of Longitude, Ruwb, and one Latitude, to find the other
Latitude 133

By one Latitude, Ruwsb, and diftance, to find the differance of Longitude
| 134
By one Latitude, Rum), and Difference of Longitude, to find the diftance

135
By one Latitude, diftance, and difference of Longitude, to find the Rumb
. 136

Cb) By

—

The Contents. | i
By the Longitude and Latiinde of two places, to find their diftance upon the
Rumb 137
By the Latitude of two places, and the diftance, to find the difference of Lon-
gitude | . 138
By one Latitude, diftance, and difference of Longitudes,to find the differences
of Latitudes s% | te ago

———

The Contents oF the third Book of the Seftor.

Chap. 1, NF the Lines of quadrature : To make a Square equal toa

Cirele, or a Circle equal toa Square Pape 141
Toreduce a Circle or a Square into an equal Pentagon, or other like fided and
_ like Angled figure. 342.

To find aright Line equal to the Circumference of a Circle, or other part

thereof | 143
Chap,2. Of the Lines of Segments: To divide a Circle into two Segments,
according to a propertion given : or to find a proportion between a Circle
and his Segments given | ; Ibid.
Chap. 3. Of the Lines of infcribed bodies : for comparing of the fides of the
five Regular bodies, with the Semidiameter of aS phere, wherein they may
be infcribed 145
Chap. 4. Of the Lines of Equated bodies, for comparing of the fides of the
five Regular bodies, with the Semidiamseter of a Sphere equal to thofe

bodies | 146
Chap. 5, Of the Lines of Metals, for finding the proportion between (everal
Metals in their weight and Magnitude ‘ Ibid,
Chap. 6. Of the Line of the leffer Tangents for defcribing of hour-lines on
feveral Planes 149
FS 1dr. S. Fotters alteration of the Settur 157

The

:

+ a

The Contents of the Firft Book of the Crofs-Staff.
CHAD. I a

=“

F the defcripsion of the Staff, and infeription of the feveral Lines

| 199
Chap.2. The nfe of the Lines of Inches, for Perpendicular heights and
diftances | |

202
Chap.3. The ufe of the Tangest Lines for taking of Angles, and obferving
the Altitude of the Sun 207

Chap.4. The ufe of the Lines of equal parts, joyned with the Lines of

Chords, for protratting of right Line Triangles 210
Chap. §.The uf of the Meridian Eine in making of a Sza-Chart, and
__ pricking down the way of a {hip : 212
Chap. 6. Of the general ufe of the Line of Nambers for finding of propor-
tional Numbers, and extrattion of Roots 216
Chap. 7. Of the general ufe of the Lines of artificial Sines 221
Chap. 8. The ufe of the Line of Artificial Tangents, in ref olving of Sphe-
rical Triangles 222
Chap. 9. Of the general ufeof the Lines of Sines and T: angents juyned with
the Line of Nunebers in refolving of right lined Triangles 224

Chap. 10. The general afe of the Line of Verfed Sines in refolving of a
Spherical Triangle, wherein three fides are known, and an Angle required
| 3 0331

The Contents of the fecond Book of the Cro/s- Staff.

Here the former Lines of proportion are more particul.rly explained

¥ V in fevcral kinds : aes 232

- Chap.t. The nfe of the Line of Numbers in faperficial meafure ae a
ras A Ler fided Superficies having any of the two fides Parallel, to find

_ the Area

Rane:
5g To find the Areaof any Triangle Ibid,
EE? To find the Areaof an Equilateral Triangle bhi 238

(b 2) =e

| The Contents.
Ee To find the Area of the Trapefia whofe fides are neisher equal nor

parallel Ibid,

£3. To find the fide of «Square equal to an Oblong Saperficies 239
¢-s— Of the Menfaration of Regnlar Polygons ; 240
¥ 2. Of the Afenfuration of Circles 241
(4. By the Diameter to find the Circumference Ibid.

| 2, By the Circumference to find the Diameter 242

‘13. By the Diameter to find the Area Ibid.

4. By the eAre: to find the Diameter Ibid.

2 5. By the Circumference tofind the Area 243

6. By the Area to find the Circumference Ibid.

7. By the Diameter to find the (ide of a Square equal tothe Cirele Ib.
8. By the Circumference to find the fide of a Square equal tothe

i _ Cirele 244.
Char.2. The wfe of the Line of Numbers in the meafure of Land by
Perches and Acres ; | . Ibid.
Chap. 3. Of the ufe of the Line of Numbers in felid meafure in finding the
content of a fauared Solid 248
And of a Cylinder 25
xa, Of the Menfuration of Cones 255
». By the Dianstter and length of the fide, to find the Saperficial
content Tid.

2. By the Diameter and Axts to find the Solid content. Ibid.

¢2, Of the Menfuration of Spheres 256
C1. By the Diameter to find the Superficial content Ibid.

2. By the Superficies to find the. Axis lbid,

== 3. By the Axts to find the Solid content 257
4. By the Solid content to find the Axws Ibid.

F25 Of the Afen{aration of Prifwms ; Ibid.
ra} c. Of Triangular Prifms Ibid,.
2. Of multangalar Prifms 258

F235 Of the Menfuration of Pyramids 259
E35 Of the Menfurationof Fruftrumss of Pyramids and Cones Ibid.
Chap. 4 The ufe of the Line of Numbers in gauging of Veffels 261
Chap. 5.Of refolving fuch Aftronomical Propujitions as are of ordinary ufe

in the prattice of Navigation, as. in finding the Altitude of the Sun

263,
The Suns Declination: the time of the Suns rifing and fetting 264.
The Amplitude | 265

The

eat The Cotitents,
The time and amplitude when the Sun cometh to be due Eaft cr Weft

266

The Suns Altitude and Azimuth at the honr of fix 267
The Azimuth at any Altitude | Ibid.
The hour of the day 270
The Right 04 /cenfton 272
With the manner of refolving thefe Propofitions by Tables of artificial Sines
and Tang: nts 277
And the finding of the variation of the Compafs ‘278
Chap 6. Of {ach nautical queftions as are of ordinary fe concerning Lon-
gitude, Latitude, Ramsb, and diftance ome 280
With an Appendix of the ufeof an Inftrament in form of a Crofs Bow, for
the more eafie finding of the Latitude at Sea 299
ES e Table of the Right Afcenfion and Declination of fome eminent
fixed Stars : 312

\. De Er
The Contents of thethird Book of the Cro/s-Staf

ae diffinttion of . Planes whereon hour lines may be deferibed

Pier

Of the nfe of the Lines.of Numbers, Sines, ana Tangents, for the drawing
6f hour-linesin all forts of Planes Ibid.
To find the inclination of a Plane. 5
To find the declination of a Plane | | 6
Chap, t. To draw the honr-lines in an EquinoBtial Plane 9
Chap, 2.70 draw the hour-linesin a direét Polar Plane IO
Chap, 3. To draw the howr-lines in a Meridian Plane 14
Chap. 4.70 draw the hour: lines in an Horizontal Plane 15
Chap. 5: To draw the hour lines in a prime Vertical Plane 1g
Chap. 6. To draw the hour-linesin a Vertical inclining Plane 2r
Chap. 7. To draw the hour-lines in a Vertical declining Plane 24.
Chap. 8, 70 draw the howr-lines in a Meridian inclining Plane 37
Chap. 9. Todraw the hour-linesin a Polar declining Plane 42

Chap. 10. 70 draw the hour-lines in a declining inclining Plane 47
Chap, 11. To defcribe the Tropicks and other Parallels of declination inan
Eqninotlial Plane | 55

~ Chap.r2,

3
7 econ

‘Chap. 15.70 defcribe the Parallels of the length of the day in any of t

; 74
Chap. 17. To draw -the hour-lines from S un-rifing, and San-fettiag, in the

The Coritents:

“Chap. 12. To defcrike the Tropicks and other Parallels of declindton ina

Polar Plane : : | iy
Chap. 13. Todefcribe the Tropicks, and other Parallels of declination in any
other Plane, not Equinottial, nor Polar 6

5

Chap.14. To defcribe the Parakels of the Signs in any of . the former
Planes | . Bah &

€

.

former Planes | bid.
Chap. 16.Z draw the old unequal Planetary hours in the former Planes

former Planes f 16
Chap. 18, To draw the Horizontal Line in the former Planes 77
Chap. 19. To draw the Vertical Circles or Azimuth in the former Planes-

79
Chap. 20. Todefcribe the Parallels of the Horizon in the former Planes

| 8§
Todefcribe {ach Lines as may (bew the proportion of the (hadow-unto the

Gnomon 95
Lafily, an Appendix concerning the ufe of a {mall portable Quadrant for
the more eafiefinding of the hour andthe Azimuth 97

Chap-1. Of the defcription of the Quadrant

Ibid.
~ Chap. 2, Of the ufe of the Quadrant in taking the Altitude of the Suny

AMaon, er Stars

113
Chap. 3. Of the Ecliptick 114
Ciap. 4. Of the Line of declination | 1g
‘Chare 5. Of the Circle of the months and days | Ibid.
Chap. 6. Of the bowr lines 117
Chap. 7. Of the Horizon | , 12%
Chap. 8. Of the five Stars 122
Chap. 9. Of the Azimuth Lines 124
Chap,10,0f the Quadrant 125.
A fecond Appendix, being the defcription and afe of a Quadrant of Afr. Bo-

{ters invention 119
TtsDefcription _ Ibid.

dts Ufe 13

A. Te find the Declinatiss Ibia.
2. To rectifie the Bead to find the Hand and Azimuth 136
3. The Afcenfional difference 137

4o The

he RIES eet Ae ae 3

eer.

The Contents, .

— 42 The Amplitude | ve Ibid.
5. The Twy Light PnP be RtL OY ay SEs.
To find the requifites inthe msoft ufwal forts of Sen Diadls === '33.9

The height of the Sun upon any Azimuth - 142

| The height at any hoar Egg
To finde The Azimuth ‘S38 Ibid.
The honr bythe Sun 146

A The declination of a Plain 147

70 make declining Dials oe ne Mags
7 0 make for Decliners ie ee 150
To make Exnf? or Weft Dials ! 154
To find the hour bythe Stars ; 1§9

_ To take Altitudes of buildings ae 161

TE ISI

The Contents of the general ufe of the Canon, and
Table of Logarithms. cs

Chap.t. | Oncerning the ufe of the Line of Nambers, 1s fet down ten
Leneral propofitions in the fe of the Crofs-fraff, and thefe
may be applied tothe Tables of Logarithms’ ~ - 166

The Uf eof the Table of Logaxithms in Arithmetick

bid.

The fe of the Canon of Signes and Tangents in the folution of Spherical
Triangles — | 182
The Ufe of the Canons of Sines and Tangents, with the Table of Logarithnes

in the folution of ‘right lined Triangles 198:

Chap. 4. Containing fome nfe of right lined T; riangles in the prattice of
Fortification

THE

207°

Advertifement, —

| Hlereas the whole Subject of the following
\ \ Treatifes do contain the ufe of Inftruments, and
that the true and exatt making of them is prin-
cipally to be minded and enquired into, I thought good to
give notice, That if any Gentlemen fiudious in the Ma-
thematicks have or {hall bave occafton for any Inftrament
belonging to this Book. as aljo with all others ufeful both
for Seaor Land, they may be furnifhed either in Silver,
Brafs, or Wood, by Walter Hayes, at the Crofs-daggers
_in Moor- fields, next door to the Pope’s-head Tavern .
where they may have all forts of Maps, Globes, Sea-plats,
Carpenters Rulesy Poft and Pocket-Dials for any Lati-
tude, &Ce

(3
ae

re

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Lie
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aaa Oe A

A ious 5
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te

Sat pe = SSESES ES PARES eR EE
50 3 Sauts

Sawy:

Gf sll ther Safeunents fr the hana rd

Wale Hes at the

| SEA Se OO ce |

(This § 7

Be ge the Seétor accor,
ie oe eS 4

———

; te the

er

=

-Saaaage

a.

THE

“FIRST BOOK

fer CT OR

The aay , the Making, and the General Ufe
of the SECTOR,

SecTor inGeometry, isa Figurecomprehended — -

of two right Lines containing an Angle ac che
Center, and of the Circumference aflumed by
chem. This Geometrical Inftrument having two
Legs, conteining all variety of Angles, and che
diftance of the Feet, reprefenting the Subtenfes
SHOE of the Circumference , is therefore called by che
SIL WS GER fame name.

It conteineth 12 feveral Lines or Scales, of which 7 are general, the
other § more particular. The firft isthe Scale of Line divided into 10°
equal parts, and numbred by 1, 2, 354) 5: 6) 7» 8:9, 10.

The fecond, the Lines of Superficies, divided into 100 unequal
and numbred by 1, 23 35 455s 6, 7; 8, 9, 10.
3, Thechird, che Linesof Solids, divided into 1000 unequal parts,

and numbred by 5,1, ¥y 2) 3) 4) $16, 7» 8,9, 10. ;
4. The fourth, the Lines of Sines and Chords, divided into 9 degrees,

and numbred with 10, 20,30 unto 9%
i Thefe

parts,

te cat Ps ee
eee

| 4 The Defeription of the Lines.

3. To divide the ines of Super ficies.

“N Ecing the Superficies do hold tn the Proportion of their himolog Al
S Sides duplicated by the 29. Prop. 6.Lib Eucid. [ft you thall find
mean Proportionals between the whele Side, and each hundred part of
the like Side, by the £3. Prop. 6. Lib. Euclid. ail of them cutting the
fame Line, thar Line fo cut fhall contain the Div fions required ; where-
fore uponthe Censer A, and Semidiameter equal to the Line of Lines?
defcribe a Semicircle AC B D, with A B perpendicular to the Diameter
CD. And letche Semidiamerer A D Le divided as che Line of Lines
inco an hundred parts, and A E the one half of AC divided alfo into
an hundred parts, fo hall the Divifions in A E be the Centers from
whence you thall defcribe the Semicircles C 10. C 20, C 30. &e divi-
ding the Line A Binto anhundred unequal! parts: and this Line AB ~
fo divided fhall be che Line of Superficies, and muft be transferred inte
the Se&tor. Butlec che numbers fetro them be only 3,1. 2. 3. unto 10,
as in the Example. 3

Or thefe Lines of Superficies may otherwife be cransferred into the
Se&tor, out of the Line of Lines, by a Table of Square Roots; For the
Root taken out of the Line of Lines, fhall give che Square in the Lines °
af Superficies. |

As, to infcribe che Divifion of 25 in the Lines of Superficies; put fix

‘Ciphers to 25, and make it 25000000, then find che Square Root of

this Number, which will be so000.

Take therefore 5000 out of the Line of Lines (fuppofing che whole
Line to be 0000) and ic will give the true Diftance between the Center,
and the points of 25, inthe Lines of Superficies. r.

So,for the Divifion of 30,putto 30 fix Ciphers,and make it 30000000,
whofe Square Root is §477+ _Fhis (caken out of the Line of Lines) thall
give the place for the Pointsof 30, inthe Lines.of Superficies. And the

like reafon holdeth for all che reft, according to this following Table,

If any pleafe to make ule of a Diagonal Scale, equal co the Line of
Lines, he may put eight Ciphers to the Number propofed, and make the
Table of Roots to five Places: So, his work will be more exaét.

A Table of Square Roots for the Divifion of the Lines of Superficies,

ie

2 peer -

a IE SS SSS
A Table of § guare Roots fun Divifi ton of the Line of Superficies,

FR, “epee OF Sp

See eS es See ce |

Sq | Root, \Sq.' Roo’, 4 Rost Sq] Root aS | Root, ae Root. | $q. | Root.
C spe 3015 477 l45/6r0 Ir75 ol75\8 ae 9487
sal pte ih ae ware : = eee
TOL} (3937 | 1952 O7 45) 477704 9 2OBY, Lo S43
1] 1O00}16 40001325, 63 4.6) 67 82! om 7810 76) 85 I$! 91) 9539
1225 4.062 yOr2 6819 785 lye 7 40 OF 65
2141417 4123 32 5657\4716556 621737 4 sto7/8e95 92) 9592
pase 41831 P72" Vc 7906 Hoe es 9018
3{°'732118)4.243133 v7.44 148 6928/63 or le 93 9644 |
1871 (4301 5788] [6962 7909) 9860 9670
42000)1914359134)583 1 4.97000 '54,3000)79 8885) 94) 9695
2421 44.16 5874] 17036 Bost 8916 9721
512 236/20/4472)/35 35]5916/50) 7O71 65 8062/80, 8544 &944/ of 0747
[ected et: £0 fet Saas fred bl i a se A Lies aes Ih aa ky RL Pe Peres:
—|
2345 4528! 5958 ;7TOCy {8095 8972 | 9772
6|244.9]2 1/45 8213616000) 51i7141166) 3124181|g000] 96] 9798
2550] |4637 6042) 7176! |8rss1 loo28) 93:23
7|2646!2 2: 4690 3716083 | 52) atl Onie ae) 82/9055! 97} 9849
oe: 14743 Ae 7246 issu 90831} 98>
812528)23, ae 38) 6164153 7280! 68 8246|83)Q110 foe 9899
2915} 4848 ee 73 E4. §276 9138 9925
89913916245! 816918307/84lo16
9{3000 244. 99 39,0245 1541734516915307 9105S; 99! 9959
3082] 4950 62 85| 7382 $337 9192 9975
10]3 462/25 | 5000/40 93250555 7416}7018367/°5/9219} t00/ 10000
— |) fae OO | Sigh ae ns | ny fe fee

|
[7450] 133961 9247

3240 sosol 0364, f
ape 26/\5O99|4! 6403|56!7483 71 18.426 86 9274

3391} 15148) |6442] |7517] 18556] 9300
12)3464/27|5196/42/6481157]7550172/848 518719327

35361 13244 Gra! 7383; 18515} 10354
13]3606/ 281 529114316557] ° {7646173/8544 18819381

3674) 5338] [o505} |7648! 18573 lo4ort
1413742|29|5385!4416633159{7681174|8602|80 94344

$808) 1543: 6671 7714] 18631 9460] :
1§|3873 Baraat. 4516708|60'774.6}7518 660100 25 e733 15477145 10705160177 4.6)75 186601001 94R7t 3 |

ior ee
4 ;

af/d Ry ae 4 a ” ye, Spt + ’ cd
4 ® fT sath hy

6 — The Defeription of the Lines:

4. Todivide the Lines of Solids.

a eeealts Solids do hold in the Proportion of their homologal Sides
triplicated, if you fhall find cwo mean Proportionals between the
whole Side and each thoufandth part of the like Side: allof them cut-
ting the fame tworight Lines, the former of thofe Lines fo cut, thall
contain the Divifions required. :

Wherefore upon the Center A, and Semidiameter equal to the Line
of Lines, deferibea Circle and divide ir into 4 equal pares C E BD,
drawing the crofs- Diameters CB ED, Then divide the Semidiameter
AC, fift into 10 equal parts, and between the whole Line A D and
AF, the centh part of AC, feek out two mean Proportional Lines
Aland AH: again berween AC and AG (being two Tenths of
AB) feek out two mean Proportionals AL and AK, and {0 for-
ward in thereft. So fhallthe Line A B, be divided into ro unequal
parts.

Secondly, divide each cemh part of the Line A C into 10 more
and between the whole Line AD, and each of them, feck out two
mean Proportionals as before: So fhall the Line A B be divided now
into amhundred unequal parts,

Thirdly,

‘Thirdly, if che Length will bear it,fubdivide the Line A C once again,
each part inten more, and between che whole Line A D, and each Sub-
divifion, feek two mean Proportionals asbefore. So fhall the Line A B
be now divided into 10co parts. But the Ruler being hore, it thall
fuffice, if thofe ro which are neareft the Center be exprefled, the reft be
underftood to be fo divided, though a€tually they be divided into no
more than § or 2, and this Line A B fo divided thall be the Line
of Solids, and muft be transferred into the Seftor: Buc lec the Num-
‘bers. fec co them be only 4. 1. 1, 2 3. Ge. unto 10, as in the
Example. ‘

Ox

8 = == “the Defeription of the Lines.

Or thefe Lines of Solids may otherwile be cransferred into the Se

Vee
beer
i€

tor, out of che Line of Lines (or ‘rather, out of a Diagonal Scale
equal co che Line of Lines) by a Table of Cubigue Roots, For the
Root taken out of the Line of Lines, thallgive the Cube in che Line

of Solids. ; :
As toinferibe the Divifion of rag in the Lines of Solids; put xii,

- Ciphers co 425, and make it 125c00000000000: Then find che

Cudique Root of the Number, which will be s0000. Take therefore
50000 out of the Line of Lines; ({uch as the whole Line is ro0@e0)
and it will give the true Diftance becween the Points of 125 in the -
Lines of Solids.

So, for the Divifion of 300, putco 300 xii. Ciphers more, and make
It 300000000000008, whofe Cubdique Root is 66943. This, taken
out ofthe Line of Lines, {Rall give che place tor the points of 300 in the
Lines of Solids. And the like reafon holdeth for all the ret) accor-
ding to the enfuing Table, “a

A Table of the Cubique Roots.

Je AT hk

19 | 2668

49 | 3659

120

ATable of Cubigne Roots.

4931 | 270

6463 | 420 | 7488

20 pete Vise 125 | $200 | 275
i C

Cub 1 Root. = Rost. | Cub. ( Root. | Cub. | Root. | Cub. | Root.
fe) O 120 4271¢ | 50 | 3684 | 125 |! S000 ! 'a7g | 6502
| | 794, | 21 eer 52 | 3732 | 130 hee 280 | 6542
£7) 1000 ; 22 | 28c2! 954 | 3779 | 135 | $129 | 285 1 6580.
1144 | 23 12843] 56 re 140 5193 | 290 | 6619

2 | 1259 | 24.) 2884} 58 | 3870 145 | 5253 295 | G656
1357 | 25 | 2924 | 60 | 3914 | I50 ca 300 | 9694}

—. § — | ——g —— ! — / | —_ }| __

3 8 | 3 [296 62 3957 | et) 5428 | | 6731
1518 | 27 } 3000 | 64 | 4000 | 160 | 5428 | 310 6767.

4. | 1987 | 28 | 3036 | 66} 4o4r | 165 Be 315 | 6804
1650 fe 3072 | 68 48k 170 5539 | 320 | 6839

- § | 1709 ie 3107 | 70 1 4t2t | 175 | 5593.) 325 | 6375
“11765 | 31 | 3142 72 | 4160 | 1801 5646 pe | 6910

6 11817 1-32 | 3174 || 74. 4198 | 185 | $608 | 335 | 0945
1866 | 33 | 3207 | 7614235 | I99 5748 | 340 | 6979.

7 | 1912 | 34.) 3239, 78] 4272 4 19s 5798 | 345 | Gor3
[1957 pee 3271 | 801} 4308 | 200 pet | 359 | 7047,

8 | 2000 | 36 | 3301.| 82 | 4344 | 29 5896 | 355 7080
Es 37 | 3332 | 84 | 4379 | 210 | 5943 360 ~7II9

g | 2080 | 38! 3361 | 86 | 4414 | 215 1 5900 365 1 7146
21171 39 | 3391 | 88 | 4447 | 220 | 6036 | 370 | 7179

10 | 2454.} 40 | 3419 | 90 | 4481 | 225 Be] ha [7211
Ep erat PMY 2p SEG Manilagey Be Seat | Pe | CS yo

Ir | 2223 | 41 | 3448 92 | 4515 | 230 6126 1 380 | 7243
12 | 2289 | 42 | 3476 94] 4540 | 239 | 6171 | 385 | 7274
13 | 2351 143 | 3503 ; 96 | 4578 | 240 | 6214 399 | 7306
14 | 2410 | 44 | 3530 | 98 | 4610 | 245 1 6257] 305 | 7337
15, | 2466 | 45 | 3556 | 100 } 4641 | 250 | 6299! 400 is
16 | 2519 | 46 | 3583 105 | 4717 - 6341 1 405 17398
17 | 2571 | 47 | 3608 | 110 | 4791 | 260 | 6382 410 | 7428
18 | 2620 , 45 | 3634 | 115 | 4862 | 265 | 6423 | 4 | a

6502 425 47518

PEN ay Asie Doe RS Rey: ALGER Oi ORR BIL Aaa Ra Puta > Set ae
wd Bt fo Avge : Dpath acu ass Mi
z ik ,

: The Divifion if the Lines ul Solids.

a a OR ee ek SS

—a
ed

~ Cub. | Root. aga Sea Root. | Cub. Row.
| 425 175 18 | 575 8315 | 725 | 8983 | pid 9564) -
ere HY ee r oF80%.

30) 7547 580 8539 | 739 | 9004 “880 | 95

9 7570 385 | 8363 ; 735 | 9024} 885 ! 9600
| 440 7605 | 590} 8387 | 740] 9045 | 890 | 90T9
44s | 76344 sos | 8410 | 749 | 9065 | S95 | 9036) —
[ees 7663 | 600 | 9434 739 | 9085 | 900 | 9054)
455 | 7601 | 605 | 8457) 755 | 9tO9 | 905 9672 |
‘460 | 7719 610 | 8480 | 760] 9125 | 910 | 9690

ie, | 7747) 615 , 8504] 765 | 9145 4 915 08 |
470,' 7774 | 620 8527 1770 Q165 201 9725 | |
475 | 7802 | 625 | 8549 es 9185 | 925 9743

480 4 7829 | 630 | 8572 | 780} 9205 | 930. 9761 |
786 | 7856 |, 635 | 8595 785 | 92241 93§ | 9778
499 | 7883 | 640, 8617 | 790} 9244| 940 | 9795 |

495 | 7910 | 645 | 86401 795 | 9263 | 945 ; 9813

500. 7937 oi Boos 800 | 9283 ; 950 : 9830
Diallo |}

soy | 7963 | 655 | 8684, 805 | 9302] o55 | 9847.

510 | 7989 | 660; 8706} 810} 9321 4 900 | 9864.

s15 | 8015 563 | 8728] 815 | 9340] 965 | 9881

520 | 8041 670 8759] 820! 93591 970 98084

sz5 | 8067 | 675 8772 | 825 | 9378 | 975 | 9915

538 | 89 5 | 680 8793 | | 9397 | 980 5932)

535 7 81rd | 685 | 8815; 835 |] 9416 | 985.1 9949

quod 8143 | 1699 | 8836 |: 840°} 0435 || 990.) 9996

545 | 8168 ( 695 | 8857) 845 | 94541 905 | 9983

559 B10 (700 8879 | 850 ath ps 10000
ssy. 8217 | 705 | 8900 | 855 | 94

560 8242 | 710} 8921 , 860 | 9509

563 | 8267, 715 | 8942 | 865) 9529 i
‘570 i201 720 | $962 | 870 9546 | |
<75 } 8315 | 725 8983 | 875 | 9564 i

pest mela ae mmm a RR Warm ORT TREC 2a EN IMIRT ADD NTEA ONT ern

The Defsription of the Lines. iP

5. To divide the Lines if Sines and Tangents on, the Side of the Seétir.

U Pon the Center Ay and Semidiameter equal to the Line of Lines,
defcribe a Semicircle ABC D, with A B, perpendicular to the
Diameter CD. Then divide the Quadrant CB, B D, each of them
into go, andfubdivide each degree into two parts: For fo if ftreighe
Lines be drawn parallel to che Diameter C.D, through thefe go, and
their Subdivifions, they fhall divide the Perpendicular AB uncqaally

into gO.

\

=a El

J

WESn
Vale SONIA
ie x
ee F

\
BY

And this A B (fo divided) thall be the Line of Sines, and muft be
transferred into the Seftor. The Number fet to them are to be 10, 20).
30, Gc. unto 90, as inthe Example. |

If now inthe point D, unto the Diameter C D, we thall raife a Per-
pendicular DE, and toit draw ftreight Lines from the Center A, through
each Degree of the Quadrant DB, thefe ftreight Lines fhall be Secants,
and this Perpendicular fo divided by them (hall be the Line of Tangents,
and muft be transferred unto the Side of the Seétor; The Number fet
co them, are fo be 10, 20, 30, ec. as in the Example.
. Ef between A and D, another ftreight Line GF be drawn parallel to
DE, itwill be divided by thofe Lines am the Center in like fort as

2 DE

12°. The Defcription of Lines. —
DE is divided, and it may ferve for a lefler Line of Tangents, to be
fet on the Edge of the Seétor. } |

If the Compaffes fhall be extended from C to each degree of the
Quadrant CB, and thofe Extents transferred into one Line (C A) chis
Line C A fo divided into 66 (or rather into 90 gr.) thall be a Line of
Chords, and may be fet on fome void place of che Seftor.

Thefe Lines of Sines and Tangents, may yet otherwife be transferred
into the Seftor out cf the Line of Lines (or rather out of a Diagonal
Scale equal tothe L'ne of Lines,) by Tables of Natural Sines and Tangents.

For the Sine of 99 gr. being equal to the whole Line of Lines of -

102000 parts, the Sine of yogr. willbe equal to so00e (half the Line
of Lines;) andthe Sine of 45 gr. equal to 70710 parts of the Line of
Lines, according to the ufual Table of natural Sines.

In like manner the Tangent of 45 gr. being equal to the whole Line

of Lines,*the Tangent of gogr. will be equal to 83910 parts of
the Line of Lines: and che Tangene of 50 gr. equal to 119175,
that is, coone Radius (or whole Line) and 19175 parts more of the
fame Line of Lines, according tothe old Table of Tangents. ‘

And (upon the fame ground) the Secant of 4o gr. will be equal to
1.30540, cthacis, one Radius and 30540 parts of the Line of Lines -
and the Secant of sogr. equalto 8.95572, and fo the reft, according
to the like Table of Secants.

The Line of Chords may alfo be divided by’help of the Table of Sines
and Line of Lines. For the double Sine of half the Ark taken. out of
the Line of Lines will giveche Chord. ~

As ifthe Ark propofed were 60gr. The half of this Ark is 30 gr,

and che Sinethereof ;0000, which being doubled, make 100000, the
whole Line of Lines, equal to a Chord of 60 gr.

So for che Chord of go gr. the half Ark is 45 degrees, and the Sine
thereof 707:0, which being doubled, make 1414240, that is, one

Radius, and 41.420 parts of the Line of Lines, equal tothe Chord of

90 gr. required. :
5. To fhew theGronnd of the Seftor.

y Ec AB, AC, reprefent the Legs of che Seftor, then feeing thefe -

two AB, A Carecqual, and their SeGtions A D, A E, alfo equal,

they fhall be cuc proportionally: and if wedraw the Lines BC, D E,

they willbe parallel by Prop.2. Lib.6. of Enclid, and {o the Triangles
| ABC

The Defeription of Lines, 13
ABC, ADE, thallbe equiangled , by reafon of che common Angle
at A, andthe equal Angles at the Bafe, and therefore fhall have the-
Sides proportional about thofe equal Angles, by Prop.4.Lib.6. of Enclid.

on <a une

The Side A D thall be to the Side A B, as the Bafis, DE, unto the
parallel Bafis B C, and by converfion A B fhall be unto AD, as BC
unto DE; and by permutation AD thali be unto DE, as AB to
BC, @&e. So thatif A D bethe fourth parc of the Side A B, then DE
flaall alfo be che fourch parc of his parallel Bafis BC. The like reafon
holdeth in all other Sections.

7. To thew the general Ufe of the Setter.

| ipa may fome Conclufions be wrought by the Setor even then

when it is fhur, by reafon that the Lines are all of one length: but
generally che Ule hereof confifts in the folution of the Golden Rule,
where three Lines being given of a known Denom'nation, a fourth
Proportional is co be found. And this Solution’ is diverfe in regard
boih ofthe Lines and of the Entrance intothe Work,

The Solution in regard of the Lines is fometimes fimple, as when the
Work is begun and ended upon the fame Lines. Sometimes itis com<
pound, as when it is begun on one kind of Lines and ertded on another.
Tc may be begun upon the Lines of Lines, and finifhed upon the Lines of

© Superficies. It may begin on the Sines, and end on the Tangents.

The Solution in regard of che Entrance into the work, may be eithet
with a Parallel, or elfe Lateral on the Side of the Seétor, 1 call it
Parallel Entrance, or entring wich a Parallel, when the two Lines of
the firft Denomination are applied in che Parallels, and chethird Line,

_ and that which is fought for, areon the fide of the Seétor: I callic lateral
Entrance, or entring en the fide of the Segtor, when the two Lines ca
tne

| 14 ey Tee general Ufe of the Settor. :
the firt Denominarion are on che fide of the Seftor, and the third
~ Line, and chat which is co be found out do ftand in the Parallels. _

As for Example, let there be given three Lines A,B, C, to which I
am cto find a fourth Proportional, let A meafured in the Line of Lines
be 40, B so, and C 60, and fuppofe the Queftion be this: If 40
Months give so pounds, what fhall 6¢? Here are Lines of two De-
fiominations, one of Months, another of Pounds, and the firft, with
which IT am co enter, muft be that of 40 Months. If then I would
enter witha Parallel, firft I cake A, the Line of 40, and pat it over as
2 Parallel in'50, reckoned in the Line of Lines, on either fide of the
Seétor from the Center, foas it may be the Bafe of an [fofcheles Tri-
angle BAC, whofe Sides A B, AC are cqualto B, the Line of the
fecond Denomination.

The;

~

| The gentral Ufe of the Settor. 15

Then the Sedtor being thus opened, I take C the Line of 60, be-

tween the Feet of the Compaffes, and carrying them parallel co BC,

I find them to crofs the Lines A. B, AC, on the fide of the Sector

in D and E, numbred wich 75, wherefore I conclude the Line AD

or A E is the fourch Proportional and the correfpondent Number 75,
which was required. : |

ue

he ee ee res gussets oa

But if Iwould enter on the Side of the Se&tor, then would I dif-
pofe the Lines of the firftDenomination A andC in the Line of Lines,
on both fides of the Sector in AB, AC, andin AD, AE, foasthey
fhould all meet in the Center A, and then taking B the Line of the
fecond Denomination, put ic over as a Parallel in BC, that ic may be
the Bafis of the Hofcheles Triangle BA C (whofe Sides AB, AC, are
equal to A che firft Line of the firft Denomination) for fo the Se@or
being thus opened, the other Parallel from D to E, thall bethe fourth
Proportional which was required, and if it be meafured with the other
Lines, it fhall be 75, as before.

In both thefe manners of Operations,the two firft Lines do ferve ro open
the Seétor to his due Angle, che Difference between them is alpen

this,

16 —— Phegeneral @[e of she Setter.
_ this, that in Parallel Entrance, the two Lines of the firft Denomination,

are placed in che Parallels BC, DE, and in Lateral Entrance they are
placed onboth Sides of che Se@tor, in AB, AD, andin A Cy, AE,

_. Now in fimple folution which is begun and ended upen the fame

kind of Lines, ic is all one which of che ewo latrer Lines be puc in the
fecond or third places. As in our Example wemay fay, 4s 40 are ta 59,
fo 6a unto 75, orelf{e, 4s 40 are to 60, {050° mato 75. And hence ic
cometh, that we may enter both with a Parallel, aad on the Sides two
manner of waysat either Earrance, and fo che mft part of Queftions
may be wrought four feveral ways, choagh in the Propofitions following,
¥ mention only chat which is moft convenient. If any have noe che Seor,
he may make'ufe cf che former Figure, as in our Example, where we
oN aa Numbers given ( 40. 50. 60,) to find che fourth Propor-
tional.

Firft, draw aright line (A D) to reprefent one of che Lines of the Se-
Gor. Then take our the firft Number (48) out of the Line of Lin
and prick ic down from A toB; and on the Center (A,) and Semi-
diameter (A B) defcribean occule Ark of a Circle from B cowards C.
In like manner, take out (60) the ocher Number of the firft Denomi-
nation, and prick icdown fromAtoD. And onthe Center (A)and
Semidiameter CA D) defcribea fecond Ark of a Circle, from D coward E,

es,

That done, take the third Number (§0) and infcribe ic into the firtt Ark

fromBtoC; and laying she Ruler tothe Center (A) and the Point C;
draw the right Line AC, out inlength, till it cutthe fecond Ark inthe
pointE. So the Diftance from D to E (taken and meafured in the
fame Scale wich che chird Number) will give 7g for the fourth Pro-
portional, | | :

Thus much for the general bife of the Se&tor, which bein

g confidered,
and well underftood, there is nothing hard in that which

followeth,

CHAP,

b
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fal
ay
lk
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a
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ie

CHAP. IIL.
The wie of the Scale of Lines:

x. Tofet down a Line, refembling any given Parts oy Fraction of Parts,

apes Lines of Lines are divided a@ually into 100 parts, but we have
A putonly ro Numbers in them. Thefe we would have to fignific
~either chemfelves alone, or cen times themfelves, or an hundred times
themfelves, or athoufand times themfelves, asche matter thall require,
As if the Numbers given be no more than 10, then we may think the
Lines only divided into ten parts according to the number {er to them,
Ifchey be more than ro, and not more than 100, then either Line thall
contain 100 parts, and the Numbers fet by them thall be in value 10,20,
30,@cs as they are divided actually. If yet they bz more than 100,
then every part muft be thought to be divided into 10, and either Line
fhallbe 1o0e parts, and the Numbers fet co them hall be in value 100,
200, 300, and fo forward {till increafing themfelves by-10, This being
prefappofed, we may number the Parts and FraStion of Parts given in
the Line of Lines; and taking ouccthe Diftance with a Pair of Come
paffes, fec it by, for the Line fo taken fhall refemble the Number gi-
ven.

In this manner may we fet down a Line refembling 75, if either we
take 75 out of the hundred parts, into which one of the Line of Lines
is actually divided, and note itin A, or 7% of the firft ro parts, and
note it inB, or only { of one of chofe hundred Parts, and note it in C,
Or if thisbe eicher too great or too {mall, we may runa Scaleat pleafure,
by opening che Compatfs to fome {mall diftance, and running it ten times
over, then opening the Compafsto thefe ten, run them over nine times
- amore, and fer Figures to them asin chis Example, and out of this we

may take what parts we will as before. |

To this end I have divided the Line of Inches on the Edge of the Sector,
fo as one Inch conteineth 8 parts, another 9, another 10, @c. according
as they are figured, and asthey are diftant fromthe other end of the
Se€tor, that fo we might have the berger Eftimate,

1B beg 2, To

‘2h

|

werner a HHA as /

i Sormncetmmmete dp nmmenretioim

A

Pap a? Bhi a 7 OES See Sued Pate Motett ke E
peaksave (as eae maee ae ig baa i Ae og his
4 ey a ed ans ¥ es She

ft Te fy
be eM
fates
RAY ane it
4 eet? sg

ROE LE |

a The Ufe of the Line of Lines =

2. To increafe a Lint ina given Proportion. i 7 7
3. To diminifh a Line ina given Proportions

““ Ake the Line given wich a pair of Compaffes, and open the Seétor,
fo asthe Feet of the Compaffes may ftand in the point of the -
Number given, then keeping the Sector at this Angle, che Parallel Di-
ftance of the points of the Number required, fhall give the Line re-
quired.

A

~ Let Abea Line given to be increafed in the Proportion of 3: to §
Firft, I takeche Line A wich the Compafles, and open the Seor till I
may put it over inthe Points of 3 and 3, fo che Parallel becween the
Points of § and5, doth give me the Line B, which was required.

In like manner, if B be a Line given to te diminithed in. the Propor-
tion of 5 to 3, I takethe Line B: and to it cpen the Seétor in the points
of 5 and g, fo the Parallel between the Points of 3 and 3, doth give me
the Line A, which was required.

If this manner of work doth not fuffice, we may multiply or divide
the Numbers given by 2, or 3, or 4, @c. And fo work by their Num-
bers ¢gui-multiplices, as for 3 and 5, we may open. the Seétor in 6 and
70) or clfein gand 15, orelfein 12and 20,orin 15 and 25, or in18

and 39, Oe.

4. To divide a Line into any numsber of Parts given. : 4

a Ake the Line given, and open the Sector according to the length of

the faid Line in the points of che parts, whereinro the Line fhould :
bedivided, chen keeping theSefor at this Angle, the Parallel Diftance 1
between che points of x and x fhall divide the Line given into the Parts .
required, f

Lee

ig a ala fd Sarah Fe nh ih ae BN Sal ON tts each ety hia kn heh ee ae coe al, oe kaa Ol eee ct
Med OES 3 vee s 5: WHS wy Sr AS y Py 7 ia 4 Soe sg ey He
ar See > i ¥ ‘ ! \S

: The Ufe of thé Line of Limes. 49
Let AB be the Line given, to bedivided into five parts, firft 1 cake
this Line A B, and to itopen the Sector in the point of 5 and 5, the Pa-

rallel becween che points of t and 1, doth givemethe Line A C, which
doth divide it into the parts required.

fo A> 20 23
A cake awe

Or let the like Line A B be to be divided into twenty chree parts. |

Firft, I take out the Line and purit upon the Seftor in the points of 27,
then may I by che former Propofition diminith it in A-C, C Dy, inthe
Proportion of 23 to 10, and after chat divide the Line A C into 10, eee
as before. . :

5- To finda Proportion between two or more right Lines given.

| 3 Bea the greater Linegiven, and according to ic open che Sector im
the pointsof 100 and 100,’ then cakethe leffer Lines feverally, and

carry them parallel co che greater, till they ftay in like points, fo the Num-~

ber of points wherein they flay, fhallfhew their Proportion unto 100.

A B
(er IE
q Ei {D>

Let the Lines givenbe AB, CD, firft I rake the Line CD, and to
it open the Sectorin the poiats of 100 and roo, then keeping the Seétos
at this Angle, I cncer che iefler Line AB, parallel tothe former, and
find it to crofsthe Lines of Lines in the points of 60. Wherefore the
Proportion of A B tu UD, is as 60 to 100,

Or ifthe Line C D be greater than can be put over in the Points of
roo, then I admit the lefler Line A B to be 100, and cutting off CE
“equal to A B, I find the Proportion of CE untoED to be as 200,
almoft 10.673 wherefore this way the Proportion of AB unto C D, is
as 100 unto almoft 167.

This Proportion may alfo not unfitly be wrought by any other Num-
ber, chat admits feveral Divifions, a namely, by the Numbers of ‘Ss
| 3 An

44s) Ye gay ena ol Sy ho ohm g aang tic 4
eS eae Dey,

£0 The Ufe of the Lines of Lines.

‘And fotheleffer Line will be found co be 36, which is.as before in leffer
Numbers, as 3 unto 5, It may alfobe wrought without opening the
‘Se€tor. For if the Lines between which we (eek a proportion, be applied
t6 the Lines of Lines (orany other Scale of equal parts) there will be
fuch Proportion found between them, as between che Lines co which

they are equal,

6. Tw0 Lines being given, to find 4 third is continual Proportion.

FQ place both the Lines given, on both fides of the SeGtor from the
, Center, and mark the cerms of their Extenfion, then take out the

4 pie

, AAS te SIS TOLL Tah Rea a DES SN
AL 18 cr

P . tse De

LN EE he eee es Ee Bey Ta ee OR i mt a” AOR ess ee IP eee? @ PS DN Gg ee ea. es S

The Vie of the Lines of Lines. ane

fecond Line again, and toit open the SeGtor, in the terms’ of the firft
Line, fo keeping the Seétor at this Angle, che parallel Diftance between

the terms.of thefecond Line, fhall be che third Proportional.

Let the two Lines given be’ A B, A Cy which I take ouc and place
on both fides of the Sedtor, fo as they all meet inthe Center A,let the terms -
of the firf€ Line be Band B, theterms of the fecond C and GC, Then
dol take our A G thefecond Line again, and to ic open the Seétor in
theterms BB, So the Parallel between C aad € doth give me the third
Line in continual Proportion. For as AB isunto AC, fo BB equal .
to AC, isunto CC.

7. Three Lines being given, to find the fourth in difcontinwal Pree

portion.

He firft Line and the third are to be placed on both fides of
the Sector from the Center, then take out the fecond Line, and’
toit open the Seétor in the cerms-of the firft Line. For fo keeping the
Sector at this Angle, che parallel Diftance berween the terms of the
third Line, fhall be che fourth Proportional, Let the three Lines gi--
ven be A, B,C. |

~

Firft, I take out A andC, and place them on both fides of the Se>._
Gor, in AB, AC, and AD, AE, laying the beginning of both Lines

at

See PP eee a Wp > ts Sr 9 Ss yy ate SB ee eT IA IA RY
PS tae a CoA! AL aeRO er aneaie a ANE REED Sh OL fap SR ater ner RSET TOS SAA Le Gah roe OED a oD
eek ape bau. Giteuiess - Vie BOTs Foncan eA Sy ane Mong, Ay eee La eS coo?

RN a, Sele ~ nating i y i 1p wae F se Fi

Racer Pee arsed . ij A «

: \

ani‘ i‘é WC Ue off the Listes of Lines:
at theCenter A, then do Ttake out B the fecond Line, according toie .
T open the Se€tor in B and C, the terms of the firftLine: fo the Paral-
lel between D and E, doth give me che fourth Proportional which was
required. | |

fa. in eArithmetick, it fafficeth if the firft and third Number given
be of one Denomination, the fecond and the fourth which is required
be of anovier. For one and the fame Denomination is not requi-
red ‘neceffarily in chem all. So im Geometry, ic {ufficeth if che Sides
AB, AD, refembling che firft and chird Lines given be meafured
in one Scale, and the Parallels BC, DE be meafured in another.
Wherefore knowing the Proportion of A the fift Line, and Cthe third -
Line by the fifth Propofition before. Which ishere as 8to 12, andde-
feending in leffer Numbers, is as 4 to 6, or as 2 to 3, or afcending into
greater Numbers, as 16 unto 24, or 18 to 27, or 20to 30, or 30 to 45,
or40 t0-60, @e, If the Se&or be opened in the points of 8 and 8, to
the quantity of B, the fecond Line given, then a Parallel berween 42
and.12, fhall give DE, chefourth Line required. So likewife if it be -
opened in 4 and 4, then a Parallel between 6 and 63 orif in 16 and
16, thena Parallel between'24 and 24 hall give chefame DE: and
fo in the reft. .

8. Todividea Line in fuck fort as another Line ss before divided.

IL NIrft, cake out cheLine given, whichis already divided, and laying
Bitton both fides of the Sector from the Center; mark how far ie
extendeth. Thentake out the fecond Line which is to be divided, and
to it open the Seétorin the cerms of the firft Line, This done, take
out the parts of the firft Line, and place them alfo on the fame fide of
the Seétor from the Center. For the Parallels taken in the terms of
thefe parts fhall be checorref{pondent parts in the Line which is to be die

-wided. |

‘Lee AB, bea Linedivided in DandE, and BC the Line whith I

_ am to divide in fuch fort, as A B is divided. | |

Firft, I take Line AB, and son it on the Lime of Lines in A B,
AC, both from the Center A, then take I out the fecond BC, and to
. acopen the Sector in B andC, the terms of the firft Line. The Se&or

‘thus opened tohis due Angle, I take out-AD and AE, the parts of
the frft Line A.B, and place chem alfo on both the fides of the Sector

AD, AE, fo the Parallel DD giveth me BF, and the Parallel EE

giveth

rm re aa ea + “ A8° 2 Gee tee ees ee, el Me _™
SRR Oe a ee eye eee LO Me eR Rs TA PR Te NS a gt | ny MRE aE Riri de COR” MOR RIE ge
Pie, pa le BF, BS oe spay ee ast ¢ etre abt eles

The Bf of the Lines of Line 2 ;

giveth B G, and the Line BC is dividedin FandG, asistheorher Line _
AB in D and E, which was required. 2 tr

i}

Tfthe Line A B were longer than one of che Sides of the Ruler, then
fhould I find what proportion it hath to his parts AD, AE, and that.

known, I may work as. before in the former Propofition.

9. Two Numbers being given, to. find a third in continual Pro-

portion.

Irft reckon the two numbers given on both fides ofthe Lines of Lines

from the Center, and mark the terms to which either of them -ex-.

cendeth, then take outa Line refembling the fecond number again, ‘and
to it open the Seftor-in the terms of the firft number, for fo keeping che

Seétor at this Angle, the Parallel Diftance between the cerms of the.
fecond lateral Number, being. meafured in_ the fame Scale, from.

whence his Parallel was taken, fhall give. the third Number Pro-
portional.

Let the two Numbers given be 18,24.. Thefe being refembled in .
Lines, the work willbe in a manner all one with that in Prop.6.. and fo. .

- the third Proportional number willbe found tobe 32,

10, Thres

>

mv i WR @-e77 C54 We ODE Wet on ae Tae 1) Le TOM ero Colom b
BA APE an ema ieee A ue acs hii a) 3a a Gail

the Ofbaf the Lined tint

to. Three Numbers being given to find a foarth in difcontinual =~ |

Proportions

THe Solution of this Propofition, isin amanner all one with that
before in Prop.7. only there may be fome difficulty in placing of the
numbers. To avoid this,we muft remember that three numbers being given,

BCR! he at

the queftion is annexed but to one, and this muft always be placed in the.

third place, chat which agrees with this third number in denomination,
fhall be che firft number, and thar which remaineth the fecond number.

This being confidered, reckon che firft and chird numbers, which are of -

"the firft Denomination on both fides of the Lines of Lines from the Center,
-and mark the terms to which either of them extendeth, then cake oura
Line refembling the fecond number, and to it open the Seétor in the terms
_ of the firft number, for fo keeping the Sector at this Angle, the parallel
‘Diftance between the terms of che third lateral Number, being meafured
in the fame Scale from whence his Parallel was taken, fhall give the fourth
‘number Proportional.

Asif a queftion were propofed in thismanner, to yards coft 8/. how
‘many yards may we buy for 22J. here the queftion is annexed to 12 :
and therefore it thall be che third number, and becaufe 8 is of the fame
denomination, it fhall be che firft number, then z0 remaining, ic muft
be the fecond number, fo will they ftand in this order, 8,10, 12. Thefe
being refembled in Lines, the work will bein a manner the fame with
that in Prop.7. and the fourth Proportional number will be found co be
45: for a8 areto1o, fo12 wnto 15. . :

And this holdeth in dire& Proportion ; where as the firft number is
to the fecond, fothe third tothe fourth. $o that if the third number
be greater than che firft, the fourth will be greater chan the fecond; or

if the chird number be lefs than the firft, che fourth will be lefs than the

fecond, but in reciprocal Proportion, commonly called the Back Rule,
where, by how much che firft number is greater than the third, fo much
the fecond will be Iefs than the fourth, or by how much the firft number
is lefs chan the third, fo much the fecond will be greater than che fourth s
the manner of working muft be contrary, that is, che SeGor isto be
opened in theterms of the third number: and the Parallel refembling
the number required, isto be found between the termas of the firft num-
ber, the reft may be obferved as before, as for example,

if

- 7g
a

NY

es

ee, ee Pree Si a ee ee ee ee ea we er SR tad Re | i ER Fe eT ak ae ee a ee TS fe Pr OE Rt A es OS ae Nd On Sl eh 8)
eae Page ot ete) NAMA et Tee pi ee o> Ay, ok Les La ee pean UF Po OR pie ee Sa)

: The Ufe of the Line of Lines 25

If twelve men would raife a Frame in ten days, in how many days woald
eight men raife the fame Frame? Here, becanf(e the fewer men would re-
qnire longer time, though the numbers be 12, 10,8, yet the fourth Propor-
tional wil be found tobe 15. ee iy

Soif 60 Yards of three quarters of a Yard in bredth wonld hang round
about aroom, andit were required to know how many Yards of half a Yard

in lredth would ferve for the fame room. The. fourth Proportional wonld

be found tobe 90. — ,

So if to make a Foot fuperficial 12 inches in bredth do require 12 inches
in length, and the bredth being 16 inches, it were required to know the
length. Here, becanfe the more bredthy the lefs length, the fourth Propor-
tional will be found to be 9. OG

Soif to make a folid Foot, a Bafe of 144 inches, require 12 inches in
height, anda Bafe givemoeing 216 inches, it were required to know how
many inches it (ball havein height. The fourth Proportional woald be fonnd
tobe 8. | |

This laft Propofition of finding ‘a fourth Proportional Number:

may be wrought alfo by the Lines of Superficies, and by the Lines
of Solids.

CHAP. III.
The Vfe of the Lines of Superficiese
t. To find a Proportion between two or more like Superficies,

Ake one of the fides of the greater Superficies given, and according

to ic open the Sector in the points of 100 and 100 in the Lines of
Superficies, then take the like fides of che leffer Swperficies feverally, and.
carry..them, parallel, to the former, cilf they. ftay in like points, fo the
number of points wherein. they ftay,, fhall thew their Propostion un
to 100, ‘oO

‘Let A and B, be the fides of like Superficies, as the fides of two.

Squares, or the Diameters of two Circles, firft Itakethe fide A, and to
it open. the Seftor inthe pointsof 100, then keeping the Seétor to this.
Angle, Tenter the leffer fide B, parallel to the: former,. and find it to.
crofs the Lines of Superficies in the points of 40, wherefore the Propor-
tion of the Superficies, whofe fideis A, to that whofe fide is B, is as 100:
unto 40, which is in leffer number as § unto 2,

This Propofition might have been wrought by 60, or any: other Num-
her chat admits feveral Divifions. It may alfo be wrought without open-.
ing the Sector, for if the fides of the Superficies given: be applied to che
Lines of Superficies, beginning always at the Center of the Seétor, there
will be fych Proportion found between them,, as between.che number of
parts whereon they fall.

| 2. To angment a Saperficies ina given Proportion.
3: To diminifh a Swperficies in.a given Proportion.

Ake the fide of the Superficies, and to it open.the Seftor in thie points.

of the numbers given ; then keeping the SeCtor, at that Aingle, the
arallel diftance becween the points of the number required, thall give
the like fide. of the Superficies required. :

Let:

.
a a Re ee Ie

< =
a
eS |.

\

\ ¥ a
Re ee eee eee

he erent
SS ee

joie

The Ufe of the Lines of Superficiess =. 7

Let A be the fide of a Square, to be augmented in the Proportion of

2co 5. Firft, Itakethe fide A, and put it over in che Lines of Super-

ficies in 2 and 23 forthe Parallel between 5 and 5, doth give me.he fide

B, on which if I fhould make a Square, it would have fuch Proportion |
to the Square of A, as 5 unto 2.

In like manner, if B were the Semidiameter of a Circle to be dimi-
nifhed in the Proportion of § unto 2, J would take out B, and put it
over in the Lines of Superficiesin 5 and 5 5 fo the Parallel between z
and 2 would giveme A; on which Semidiameter if I fhould make a
Circle, it would be lefsthan the Circle made upon the Semidiamecer B,
in {uch Proportion as 2 1s lefs than 5.

For variety of work, the like caution may be here obferved to that
which we gave inthe third Propofition of Lines.

4. To add one like Superficies to another,
5. Yofubtra&t one like Superficies from another.

Irft, chg Proportion between like fides of the Superficies given, is to
SD be found by the firft Propofition of Superficies, then add or fubtra&
che numbers of chofe Proportions, and accordingly augment or diminifh

by che former Propofition.

As if A and Bwere che fide of two Squares, and it were required to
makea third Square equal tothem both. Firft the Proportion between
the Squares of A and B, would be found to be as 100 unto 40, or in the
leffer numbersas§ to 2; thenbecaufe 5 and 2added do make 7, I aug-
ment the fide A inthe Proportion of 5 to 7, andit will produce the fide
C, onwhich if I make aSquare, it will be equalto both the Squares of
A and B, which was required.

‘ In like manner A and B being the fides of two Squares, if it were
required to {ubtra& the Square of B, out of the Square of A, and to
make a Square equal to the Remainder, here the Proportion being as 5
to 2, becaufe 2 taken out of 5, the Remainder is 3, I would diminifh

the fide Ain the Proportion of 5 to 3, andfol fhould produce the fide
E 2 ~ Dy, on

th eA "CR ae ate, VAAN Ure Ve ee ee Ce eee Soe are Ss on Se ORS ie a wisn
Pe oe ers ars st SOLAN FS Lo WERT Oe tis sen RES 6c (a ie ge eee A
ae ¥ = : 5a Se ‘J ihe de i Wh va f 4

28 «—-—ss«eThe Ufe of the Lines of Superficies.

D, on which if I ‘make a Square, it will be equal co che Remainder,
when the Square of Bis taken out of the Square of A, that is, the two
Squares made upon B aad D, fhall be equal to-the firlt Square made
upon the fide A. : |

6. Té find a mean Proportional between two Lines given.

TJ {t find: whac Proportion is between the Lines given, as they are

Lines, by the fitth Propofition of Lines, then-open. the. Sector in

the Lines of Superficies, according to his Number, to the quantity of

theone, anda Parallel caken between the points of che Number belong-
ing co the other Line fhall.be che mean Proportional.

2/8

B *

Let the Lines given'be A andC. The Proportion between them Gs
they are Lines) willbe found, by the fifth Propofition of Lines, to be
as4to 9. Wherefore, Etake the Line C, and pucit over to the Lines
of Superficies between g and 9, and keeping the Sector at this Angle
his Parallel between 4,and 4 doth give me B, for the mean Pronarcienale
Then for proof of the Operation I may. take this Line B,, and put over
_ between 9:and g:: fo-his Parallel becween.4 and 4, fhallgive me the fir}
Line A. Whereby ic is plain, chat thefe three Lipes do hold in conti-
nual Proportion ; and therefore B.is a mean Proportional between A and
€, the extremes given. » : Fs

Upon the finding oat of this mean Proportion,. depend many Cc~

rollaries, as
To make a Square equal to a Superficies givens

F the Superficies given. be a. rectangle Parallelogram, a mean Propory
| I tional between the two unequal:Sides hall be the Side ef lid eeu
Square: . ) 4
~ Hit thall-be a Triangle, a-mean Proportion between the Perpendi
and half the Bate fhall be the Side of his. equal Square,. If Fo aulbete
any. other right-lined Figure, i¢ may be scfolved into Triangles, and fo
a

ce
-
ey

=

- ere we

The Ufe of the Lines of Superficies. 29

‘a Side of a Square found equal eo every Triangle, and thefe being re-

duced into one equal Square, ic fhall be equal to the whole right-lined

_ Figure given.

To find a Proportion between Superficies, though they be unlike one te
the other. }

ie co every Superficies we find the fide of his equal Square, the Pro-
. portion between thefe Squares hall. be the Proportion between the.

Superficies given.

Let the Superficies given be the ebfong A, and the Triangle B; Furf®
berween the unequal Sides of A, I find a mean Proportional, and note-
it inC: Thisis the fideofa Square equalunto A. Then between the-
Perpendicular cf B; and half hisBafe, I finda mean Proportional, and’
note icin B: this is the fide of a Spuare equal toB.:- but the Propor-

— tion beeween the Squares of C and By will be found, by she firft Pro-.

pofition of Superficies to beas 5 to4.: and therefore this 1s. the Propor-
tion berween thofe given: Superficies.. |

To make a-Superficies, like to one Superficies,. and equal to another.

1a the one Superficies given be the Triangle A, andthe other the-

Rhomboides B; and let it be required to make. another. Rhom--

hoides.liketo B, and equalto the Triangle A.. ce
Nis | | Firs,

30 2 ———té‘“«ST Uf Of thee Lanes off Susperrficéese — |
, Firft, between che Perpendicular and the Bafe of B, I find a mean
Proportional,and note it in B,as the fide of his equal Square, then between
the Perpendicular of the Triangle A, and half his Bafe, I find a mean Pro-
portional,and note it in A,as the fide of his equal Square. Wherefore now
as the fide Bistoche fide A, fo fhall che fides of the Rhomboides given

be to C and D, the fides of the Rhomboides required, and his Perpen-
dicular alfo toE, the Perpendicular required.

Having the Sides and che Perpendicular, Imay frame the Rhom-
boidesup, and it will be equal to the Triangle A.

Ifthe Superficies given had been any other right-lined Figures, they
might have been refolved into Triangles and then brought into Squares
as before. |

Many fuch Corollaries might have been annexed, but che means of

sees Wise Proportional being known, they all follow of chem«
felves.

7 To find a mean Proportional between two Numbers given,

Irft, reckon the two Numbers given on both fides of the Lines of

A Superficies, from the Center, and mark the terms whereunto they

extend ; then takea Line out of the Line of Lines, or any other Scale
of equal parts refembling one of thofe Numbers given, and pucit over in
che terms of his like Number in the Lines of Superficies ; for fo keepin
the Seftor at this Angle, the Parallel taken from the terms of the oie
Number and meafured in che fame Scale from which the other Parallel
was taken, fhall here thew the Mean Proportional which was required. ~

Let the Numbers given be 4and 9, If I thall cake the Line Ain
; the

«

«The Ofe of the Lines of saperficies. Zr
the Diagram of the fixch Propofition refembling 4, in a Scale of equal
parts, afidto it open. che Seétor in che terms of 4 and 4, in the Lines of |
Superficies, his Parallel between 9 and 9- doth give me B for the Mean
Proportional, And this meafured in che Scale of equal parts doth ex-
rend to 6, which is the Mean Proportional Number between 4 and 9«_
Foras4to 6, fo 6to 9. .

In like manner, if Erake the Line C,refembling 9, ina Scale of equal
parts, and toit open theSector in the terms of gand 9, inthe Lines of
Superficies, his’Parallel between 4.and 4 doth giveme the {ame Line B,
which will proveto be 6, asbefore, if it be meafured in the fame Scale
whence C wastaken. aaa

For the Figures 1, 2, 3, 4,@¢. here fet down upon the Line, do fome-

_time fignifie themfelves alone: fometime 10, 20, 30) 40, Gc. fome-
time 10, 200, 300, 400, @c. and fo ferward, as the matter thall:
require. The firft Figure of every Number is. alway that which js here:
fer down: the reft muft be fupplied according to the nature of the Que-
ftion. | ;
If you fuppofe Pricks under. the Number given (as in. Arichmetical?
Extraction) and the laft' Prick tothe left hand fhall fall under the laft:
Figure (which will be asoft as there be odd Figures); the unite will be-
beft placed at 1, in the middle of the Line; fo the Root and: the Square
will both fall forward, toward. the end of the Line. Bur,. if the laft:
Prick fhall fall under ehe faft Figure but one (which will be as ofc as
there bz even Figures) then the unite may be placed ac 1. in the beginning.
of the Line, and the Square inthe fecond length: or the unite may be.
placed at 10, in the end of the Line, fo che Root and. the Square will.
both fall backward, toward'the middle of che Line..

8. To find the Square Root of a Number.
9. The Root being given, to find.the Square Number of that Roots.

‘N the Extra€tion of aSquare Root it isufital co {ec Pricks under the
I firft Figure, the third, che fifth, the feventh, and fo forward, be-
ginning from the right hand toward che left, andias many Pricks.as fall:
tobe under che Square Namber given, fo many Figures fhall: be in che:
Root :.. fo chat if the Number given be lefs than. r00,. the root. fhall be:
only, of one Figure, if lefs than 10900, ic fhall be but two Figures 3. if -
lefs- than 100000; it fhall be three F* ures, Cres | |

Thereupon che Lines of Superficies are divided firft. into an Inandred: |

parts,

>To aly Coon Viet RD ee eC SEN Se ol Ta
we Ment z
Aer: ‘
Ma ‘
cet
i

ge The Ufe of the Lines of Superficiess

parts, and if the Number given be greater than 100, the firft Divifion-

BM ye IN) 28 TAS pee ME EE TPT ttn SR SR i et Per OE NCI a, em Fy ioe ee RUS ag tee Say
Minted Laas Lave si RG i Pos Bee a aah Ue aE
ater ast A

(which before did fignifie only one) muft fignifie 100, and the whole

Line fhall be roooo parts: if yer the number given be greater chan

1e009, the firft Divifion muft now fignifie 10060; and the whole
Line be efteemed at 1000060 paris: and if this be too little co exprefs.

#he Number given, as oft as we have recourfe tothe beginning, the whole
Line (hall increafe icfelf an hundred times.

By thefe means if the laft Prick to che left hand thall fall under the

laft Figure, which will be as oft as there be odd Figures, the Number
given fhall fall out between the Cenzer of the Se@tor and the tenth.

Divifions buc if the faft Prick thall fall under the laft Figure but. —

one, which willbe as oft as there be even Figures, then che Num-

_bergiven fhall fall our berween the tenth Divifion and the end of - the.

Seftor. 7

This being confidered, when a Number is given, and the Square
Root is required, take a pair of Compaffes, and fetting ene Foot in the
Center, excend the other to the term of the number given in one of the
Lines of Superficies; for chis Diftance applied to .one of the Lines
of Lines, fhall thew whac the Square Root is, without opening the.
Sector. ; :

Thus 36 doth give a Root of 6 ; and 360, a Root of almoft Ig: and

3600, a Root of 60; and 36000, aRootof 189, ee. A

In like manner, the neareft Root of 725 is here found to be (a-
bout) 27, the neareft Root of 7250, about 85: the neareft of 72500,

about 269: and the neareft Root of 725@00, about 851: And foin -

the reft.

On the contrary, a Number given may be fquared, if firft we extend
the Comapafies to the Number given in the Lines of Lines, and then
apply that Diftance to the Lines of Su
mer Examples.

- 10, Three Numbers being given, to find the
Proportion.

iv is plain (by Euclid. Lib.6. Prop.t9 & 20.) that like Superficies doth
hold ina duplicated Proportion of heir homologal Sides, whereupon
a queftion being moved concerning Superficies and their Sides: To

wfaal in Arichmetick, chat the Proportion be firft duplicated before the
Queftion be refolved, which is not neceflary in the Life of the Seétor,

only

perficies, as may appear by the for-

fourth in a-duplicated

It is’

*
RN ll A ee =

— : \
ee oo eee ee. Poe

The Vfe of the Line of Superficies, “33
‘only the Numbers which do fignifie Superficies, muft be reckoned in the
Lines of Superficies, and they which fignifie che Sides of Superficies, in
the Lines of Lines, after this manner. _ 7

If a Queftion be made concerning a Superficies, the two Numbers of
the firft Denomination muft be reckoned in che Lines of Lines: and the
Seétor opened in the terms of the firft Number to the quantity of a Line
out of the Scale of Superficies refembling che fecond Number ; fo his Pa-
rallels taken between the terms of the third Number, being meafured in
the fame Scale of Superficies, fhall give che Superficial Number which
was required. ,

Asita Square, whofe fideis 40 Perches in length, fhall contain re
Acresin the Superficies, and it be required to know how many Acres the
Square {hould contain, whofe fide is 60 Perches. |

Here if I took 10 our cf the Line of Superficies, and put it over in 405
in the Lines of Lines, his Parallel between 60 and 60, meafured in the
Line of Superficies, would be 223, and fuch is the number of Acres
required. For Squares do hold ina duplicated Proportion of their fidess
wherefore when the Proportion of their fides is as 4 t06, and’4 multi-
plied into 4 become 16, and 6 multiplied into 6 become 36, the Pro-
portion of their Squares fhall be as 16 to 36, apd {uch is the Proportion
Of 10 to 223,

If a Field meafured with a Statute Perch of 164 foot, thall contain
288 Acres, and it be required to know how many Acres it would contain
if ic were meafured with a Woodland Perch of 18 foor.

Here becaufe the Proportional is reciprocal, if I took 288 out of the
Line of Superficies, -and put ic over in 18 in che Lines of Lines, his Paral-
lel between t6 3 and 16 3 meafured in the Line of Superficies, would be
242; and fuch is the Number of Acres required, | :

For feeing the-Proportion of the Sides is as 16% to 18 > oY in
leffer Numbers as 11 to 12, and chat 11 multiplied into 12 become
121, and 12 into 12 become 144, the Proportion of thefe Super-
ficies thall be as 121 to 144, and fo have 288.to 242, in reciprocal
Proportion.

On the contrary, if a queftion be propofed concerning the Side of a
Superficies, the cwo Numbers of the firft Denomination muft be rece
koned in the Lines of Superficies, and the Seétor opened in the terms of
the firft Number to the quantity of a Line, out of che Line of Lines or
fome Scale of equal parts, refembling the fecond Number; fo his Paral-

‘tel taken between the terms of the third Number being meafured in the
F {ame

; 34 The Ufe of the Line of Superficies. Leet g

‘fame Seale wich the econd’ Number, hall give che fourth Number re-

- AsifaBield contained 288 Acreswhen it was meafured with a S:a-

tute Perch of 16 3, and being meafured with anocher Perch,was found co

contain 242 Acres, i¢ were required to know what was the length of the
Perch with whiclrit wwas.fo meafured, |

Here becaufe the Proportion is reciprocal, if I cook 164 out of the
the Line of Lines, and pur icover in 242 in the Lines of Superficies, his
Parallel between 288 and 288, being meafured in the Line of Lines,
would be 18, and fuch is the length of the Perch (in Feet) wherewith the
Field was laft meafured. «

For feeing the Proportion of the Acres is as 288 unto 242, orin
the leaft Numbers, as 144 10 121, and thac the Root of 144 1s 12,
and the Root of r24is at, theProportion of Roots, and confequently
of the Perches, fhall beas 12 10 11, and foaret6 % to 78 in réciprocal
>soportion.

If 360 men were to be fet in form of a long Square, whofe Sides
fhall have the Proportion of 5 to 85 and ir were required co know the
Number of mento be placed in front and file: Ifthe Sides were on'y
5 and 8, there fhould be but 40 mens buc there are 360: therefore,
working as before, I find thar, |

43 40 to the Square of S:
So 360 tothe Square of 15.

As 40 to the Square of 3:
So 360 tothe Square of 24.
and fo1s and 24 are the Sides required.

If 1000 mena were lodged in a fquare ground whofe Side were 60
paces, and it were required to know the Side of the Square wherein 5000
might befo lodged, here working as before, I fhould find thar,

As 1000, are to the Square of 60:

So 5000 to the Square of 134.
And fuch, very acar, isthe Number of paces required.

Al. Hoy.

Ag

:

|
|
|
a

—

—§ the Vfe of the Lines of Saperficies: eS

se rr. How to defertbe a Parabola, by help of the Line of Lines and Si:

perficiese s

Pon An as the Diameter, prick
down, by the Line of Lines, the
equal Parts Ao, Au, Ay, Al, Am,
An, @c. and from thefe Points raife
the Perpendiculars ox, UZ> YB» Ip,
mq, nh, &e. And upon the Perpen-
dicular ox, affume the Point x, and open
che SeStor in the Line of Superficies, {0
that ox (being the firft Perpendicular )
may fall in with the Points 1... & (the
firft of the Line of Superficies: ) Then
if you take off from the fame Line 20-2,
you fhall prick down uz, and 3.3
pives ye; and4e4y lps Se 5,m95
6 0.0, nhs Cre.
_ Or, you may begin your work from nh, which (becaufe ic is
the fixth Perpendicular ) take from nto h, the Point affumed, and
{ec that length in the Line of Superficies from .6 to 6, {fo may you
rick down the other Points correfpondently. |
Through thefe Points h, 4, ps g» with an even hand draw the

Parabola.

And here note, that Parabola’s may be defcribed’ of infinite Varic-
ties, according to the Cones from whenee they aretaken, yet keep-
ing all one and the fame lengtli.

B 4 CHAP,

3@

CHAP. IV.
The Ufe of the Lines of Solids.
i. To find a Proportion between two or more like Solids

N the Sphere, in regular, parallel, and other like bodies, whofe Sides:

l next the equal Angles are proportional, the work is in a manner the

fame, with that in the firft Propoficion of Superficiesybur that itis Wrought
on other Lines.

“Take one of the fides of the greater Solid, and according to it open the

SeCtor in che points of roooand. 1000, in the Lines of Solids, then take.

the like Sides of the leffer Solids feverally, and carry them parallel to the

former, till they ftay in like points, fo the number of points wherein they
ftay (hall thew cheir proportion to 1000, .

AL

*

Let A and B be the like Sides of like Sol
Semidiameters of two S
Firft I cake the fide A, and to ito
shen keeping the SeGor ac this Angle, I enter the
ahe former, and find ic'to crofsthe Line of Solid
and fuch is the Proportion betwee: the Solids rt

ids, either the Diameters or:
pheres, or the fides of two Cubes or other like,

pen the Se€tor in the points of 1009,
lefler Side B parallel co
sin the points of 400,

é equired, which in leffer.
Number is, as § to 2, |

This Propofition might have been wrought by 60, or any other Num-
ber thar admits feveral Divifjons,

Icmay allo be wrought without opening che Se@tor, for if the fides of.
the Solids given beapplied to the Lines of Solids, beginning always at ihe
Center of the Se&or, there will be firch Proportion between them, as bea.

tween the Numbers of parts whereon they fall. ;

2..T@.

The Ufeof the Lines of Solidss =

2. To augment a Solid in a given Proportion.
3. Todimini(h a Solid in a given Proportion.

Ake the fide of the Solid given, and to it open. the Seétor, in the
points of che Number given: then keeping. the Seftor at that Anele,
the parallel Diftance between the points of the Number required, {hail.
give the like Side.of the Solid required.» » :
If ic be a parallelopipedon: or fome irregular Solid, the other like
Sides may be found out in the fame manner, and with chem the Solids.
required, may-be made up with the fame Angles.

B— / 3
ree eran ees a:

Let A bethe Gdeof aCube, to be augmented in the Proportion of 2:
tog. Firft, I take the fide A, and pucit over in the Lines of. Solids in
2 and 2, fo the Parallel between.3-and 3, doth give me the fide B, on:
which if I make a Cube,. it will have fuch Proportion.to the Cube of A,
as. 3 tO 2+

in like manner, if B were the Diameter of aSphere, to be diminifled
in the proportion of 3t02, 1 would rake out B; and put: it over in,
the Lines of Solids,,in 3 and 3, fo the Parallel berween 2 and 2, would

‘ve me A: towhich Diameter if I fhould make a Sphere, . it would be
le(s than the Sphere, whofe Diameter: is. B, in fuch proportion as 2:is
lefs chan 3.

Here alfo for variety of work, may thelike caution be obferved to that:

which we gave in the third Propofition of Lines.

4, To add one like Solid to another. |
5. To fubtratt one like 8 olid from anothers.

“\Irft the Proportion between the fides of the like Solids givens is tobe:
| by the firft Propoficion of Solids : then add or fubtract thofe-
Proportions, and accordingly augment or diminifh by the. former Pro--
pofition. | | .
. Asif A and B were the fades of two Cubes, and it were. required to»
make a third Cube equal tothem both: firft the Proportion between the -
fides A and B, would be found co beas 100 to. 4.0, or in lefler terms as.

330. Ss he Ufe of the Lines of Solids:
§ to 2+ then becaufe g and 2 being added do makez, Iaygment the fide

A, inthe proporties of §to7, and produce the fide C, on which if I

_ makea Cube, ic will be equal co both the Cubes of A and B, which
was required. 4 3

Inlike manner A and B being the fides of ewo Cubes, if it were fe-
quired to fubtraét the Cube of B out of the Cube of A, and to make a
Cube equal to the Remainder. Here the Proportion being as §.to 2, be-
~ eaufe 2 taken out of §, the Remainder is 3, I fhould diminith the fide

A in the proportion of 5 toz, and fo I fhould have the fide D, on which
if I make a Cube, it will be equal to the Remainder, when the Cube of
B is taken out of the Cube of A, that is, the cwo Cubes made upon B and
D hall be equal to the firft Cube made upon the fide A.

6. Tofind two mean proportional Lines between two extremse Lines given.

| ee I find what Proportion is beeween the two extreme Lines given,
as they are Lines, by che fifth Propoficion of Lines, then open the
Sector in the Lines of Solids, co the quantity of the former Extreme, and
a Parallel between the points of che number belonging to che other Ex-
creme, fhall be chat mean Proportional, which is next che former Ex-
treme. Thisdone, open the Sector again to this mean Proportional in
the points of the former Extreme, and the parallel Diftance between

the points of the latter Extreme, fhall be the other mean Proportional
sequired,

LT. es 1
A B 1%
Che ia Ao - :

rE atSu ta cba wad © 2 ee

> Ler the ewo extreme Lines given be Aand D, the Proportion between
them, as they are Lines, will be'found to be as 27 co 8. Wherefore I
takethe Line Ay and putic overin the Lines of Solids between 29 and

27, and

aa

The Vie of the Lines of Solids, (39°
27, and keeping the Sector at this Angle, his Parallel between 8 and 8
doth give me B the mean Proportional next unto A, Then puc I over chis
Line B, becween the aforefaid 27 and 27, and this Parallel between 8
and § ae give me the Line C, che ocher mean Proportional which was
required, )

oe for proof of the operations I put over this Line C in the afore-
faid 29 and 27, and his Parallel becween 8 and 8 doth give me the very
Line D: whereby tr ts plain chat che e four Lines do hold in continual.
Proportion 5. and fo B and C are found tobe the Mean Proportionals be-
tween A and D che Excreme given. i

7. To find two mean proporti:nal Nambirs between two extreme Num-

bers given.

»
Irft reckon the Numbers given on both fides of the Lines of Solids,

beginning from the Center, and marking the cerms whereto they —
extend: then takea Line out of the Line of Lines, or any other Scale
of equal parts refembling the former of thofe Numbers, and pucit ver
in the Lines of Solids, between che points cf hislike Number, anda Pa»
rallel between the points belonging to the other Excreme, meafured inthe
Scale from whence the ozher Parallel was taken, hall give that mean pro-
portional Number which is next the former Extreme. This done open:
the Seétor again ro this mean Proportional in the Points of the former
Extreme, and the parallel Diftance between the points of the latter Ex--
treme, meafured in the fame Scale as before. fhall there thew the other.

mean Proportional required.

A 2.7
~——__——__——_—- are Bt weep eey ALE Rt Tee Pe
8 a La ,
ee ey Pe piceh at / sein Win baie MH
} 48

Br

Let the two extreme Numbers given be 27, and 8, if I hall take the
Line A, refembling 27 ina Scale of equal parts, and to it open the Seftor’
in 27 and 27, in the Line of Solids, his Parallel between 8 and eo
give me B, for his next mean Proportional, and this: meafured in the
former Scale doth extend to 18. Then put I over this Line B, between:
“the fore(aid 27 and 275 and his Parallel between Sand 3 doth: give: me

C, forthe other mean Proportional, and this meafured in the format
| ; Seale.

BO = The Ufe of the Lines of Solids. 3
Scale doth extend to12, Again, for proof ofmy work, I put over this:
Line C, between 27 and 27, as before, and his Parallel between 8 and
doth giveme D, which meafured in the former Scale doth extend to 8,
which was the lattcr extreme Number given 5 whereby ir is plain, chat —
thefe four Numbers do hold in continual Proportion: and therefore
-'and12 are Mean Proportionals between 27 and 8, which was re-
wired. -
; If you fuppofe ’ Pricks under the Number given as in Arithmerical Ex2
traétion,and that laft Prick to the left hand fhall fall under the laft figure,
-asin 1728, the unice will be left placed at 1, inthe middle of che Line,
and the Root, Square and Cube willall fall forward coward che end of
the Line. .

If che faft Prick thall fall under che faft Figure but one, asin 17280;
the unite may be placed at r, in the beginning of the Line, and che Cube
in the fecond length: or the unite may be placed at 0, inthe end of the
Line, and the Cube in the firft length. :

But if the Laft Prick fhall fall on the laft Figure but CWO, as in

- 1728003 then, place the unite always ac 20 inthe end of the Line:
io the Root, Square and Cube will all fall backward and be found in the
fecond length.

3. Yo find the Cabigque Root of a Number.
9. The Root being given to find the (ube Number of that Root.

iD the Extraction of a Cubique Root, iris ufual to fet Pricks under the
firft Figure, the fourth, the feventh and eenth, and {0 forward omit=
ting two, and prickingthe third from theright hand toward the lefrs
and as many Pricks as fall to be under the Cubique Numbers, fo many
Figures fhall beinthe Root. So that if the Number given be lefs chan
ooo, the Root fhallbe only of one Figures if lefs than 10C0009, it
fhall be bur of two Figures; if above thefe, and lefs than ro00000000
it (hall be but three Figures, ge. whereupon the Lincs of Solids are dic
vided, firft into 1000 parts, and if the Numbers given be grearer than
“4000 the firft Divifion (which before did fignifie only one) muft fig nifie
1000, and the whole Line fhall be rooooce: if yer che Number given
be greater than To0e000, the firft Divifion muft sow fignifie too0Ca0,
and the whole Line be efteemed at 1oc@@00C00 parts, and if thefe be
too little to exprefs the Numbers given, as oft as we have recourfe to the
beginning, the whole Line fhall increafe itlelf athoufand times,

By

© the Ufe of the Lines of Solids. 7

By chefe means, if che laft Prick, to the left hand, (hall fall under
the laft Figure, the Number given fhall be reckoned at the beginning of
the Lines of Solidsfrom 1 to 10, and the firft Figure of the Root thall
bealways either ror 2, If the laft Prick fhall fall under the laft Figure
but one, then the Number given.fhall be reckoned in the middle of the
Line of Solids, between 10 and 100, and the firft Figure of the Root
fhall be always either 2, or 3,or4. But if che laft Prick thall fallunder
thelaft Figure but two, thenthe Number given hall be reckoned at the
end of the Line of Solids, between 100 and r0G0, :

This betng confidered, when a Number is given, and the Cubique

Root required, fet one Foot of the Compaffes in the Center of the SeGtor,
extend the other inthe Line of Solids to the Points of the Number gi-
ven: Forthis Diftance applied to oneof the Lines of Lines, hall thew
what the Cubique Root is, without opening the SeGor.

So the neareft Root of 8490000, isabout 204,
The neareft Root of 84900000, is about 439.
The neareft Root of 84go000000, isabour 947.

On the contrary, a Number may be cubed, if firft we extend the
Compafies to the Number given, in the Line of Lines, and then apply
the Diftance to the Lines of Solids, as may appear by the former Ex-

- amples,

10. Three Numbers being given, to find a fourth in a triplicated Pro-
portion. 3

BANG like Superficies do holdin a duplicated Proportion, fo like Solids
A in atriplicared Proportion of their homologal Sides: and there-
fore the fame Work is to be obferved here on the Lines of Solids,
as before in the Lines of Superficies, as may appear by thefe two Ex-

amples.

be required to know the weight of a Cube whole fides is 7 inches; here
the Proportion would be,.

;

As 4.are toa Cube of 7:
So7toa Cube of 37%.

ung And

If a Cube whofe fide is 4. inches, fhall be 7 pound weight, ay if ie

Rly
‘ oct a
~ Lye
eS
?
x

BPG 0, FAS ARa rR A gie, haestey PNT eb, a

42 The Gfe of the Lines of Solids. Mee
And iff took 7 out of the Lines of Solids, and put ic over in 4
and 4, in the Lines of Lines, his Parallel between 7 and 7, meafu- P

red in the Lines of Solids, would be 37% 5 and fuch is the weight.

required. 4) | pte
If a Bullet of 27 pound weighc, have a Diameter of 6 inches, and it |

~ be required to know che Diameter of the like Bullet, whofe weight is
125 pounds; here che Proportion would be,

AAs the Cubique Reot of 27, us «nto,
So the Culigne Root of 125, 1 untoto.

And if I took 6 out of the Line of Lines, and put it over in 27, and

27 of the Lines of Solids, his Parallel berween 125 and 125 meafured

inthe Line of Lines, would be 10; and fuch is the length of the Di-
-amecer required, ;

The End of the fir Book.

THE

gt Te a RS BS Wes Urea ie rele Dae eee bi, pene WAR GP a ey Sr lyk (Ae SNe! ARUN ea eR pica Ber, Fi

ek a a Ro eA a ane. SC Lan | Vt eR EY Cam by eI ae RI Ae Ee ye
en Teh: bene ee : & any j eas

4 4

peneenenensata’
- SECOND BOOK
fork CT ©

Containing the Ufe of the Circular Lines. —

“ PhAnARNets

-

CHAP. I.

Of the Nature of Sines, Chords, Tangents, and Secants,
fit to be known before-hand , in reference to right-lined
Triangles. :

N the Canon of Triangles, a Circle is commonly divided into
| 360 Degrees, each Degree inco 60 Minutes, each Minuce into 60
. Seconds. : » ;
A Semicircle therefore is an Ark of 180 gr.
A Quadrant is an Ark of go gr.
\ The meafure of an Anple is the Ark of a Circle defcribed out of the
anpular point, intercepted between the Sides fufficiently produced.
Sothe meafure of aright Angle isalwaysan Ark of gogr. and in
this Example the meafure of the Angle B AD in the following Fi-
gure, is the Ark BC of 40gr. the meafure of the Angle B A Gy is the |
Ark BF of 5@gr. )
Fhe Complement of an Ark or of an Angle doth commonly fignifie
the Ark which’ the given Ark doth want of 90 gr, and fo the Ark
B-Piis che'Complemeiit of the Ark BC, and the Angle B AF, whofe:
| | G 2 meafure

GS Uhh e AS ee REST NEE OR MRT SUITES MAINA ESR NE Ay UMP EOP CUD Soe On ELA Nth gu BR Mun CH ae an

TNS RBS SRY NR omg be Thea MACRO are age wate aN
‘ POL PSS ( (eye a bi ts ay Bei
4 - ' | -

44 «Of the nature of Sines and Tangents. by
fneafure is BF, is the Complement of the Angle BAC; and on the —
contrary, |

| The Complement of an Ark or Angle in regard of a-Semicircle, 1s
that Ark which the given Ark wanted to makeup 180¢r -and fo the
Angle B AH is the Complement of the Angle E AF, asthe Ark EH
is the Complement of the Ark FE, in which the Ark C E is the excefs
above the Quadrant. |

“COCRAS LERCH Oka sR ota SES Seanase-

uo FO 40 “So 0 $98
The Proportions which thefe Arks (being the meafures of Angles) have:
7 to the Sides of a Triangle, cannot be certain, unlefschat which js crooked:
be brought to a ftreiglic Line, and chat may be done by the application
| of Chords, Right Sines, Verfed Sines, Tangents and Secants to the Semi-
diameter of a Circle. 3 | |
A Chord is aright Line fubtending an Ark: fo BE is the Chord of
the Ark BCE, and B.F a Chord ofthe Ark B4oF.
Aright Sine is half the Chord of the double Ark, viz. the right
‘Line which fallech perpendicularly from. the one Extreme of t

he gi«
ven. Ark, uponthe Diameter drawn tothe other Extreme of Fetal
Ark. fi i

io F So if the given Ark be BC, or the given Au le be BAC, let the
ee Diameter be drawn through the Cenrer A- unto C and a cae
bos B Drbe let down from the Extreme B upon A C, this Perpendicular B D’

fall. be the right Sine both of the Ark B.C, and alfo of the Angl

i BAC:

hie .

c

ne Cee NS ee te PaO kel Maen eke re PM NRT ee SEU Nem ey Cr OR: eas Ree epee ley, el ee
ts mn Ee i he ee ee ‘ 7 ene . ve mse * Nt in is £ " . =o

7
ha

Of the nature of Sines and Tangents. 4y

B AC: and it is alfo che half of the Chord BE, fubtending the Ark

~ BCE, which is double to the given Ark BC. Tn like manner, the Se-

midiameter FA, is.the right Sine of the Ark FC, and of che righe

- Angle FAC; for ir falleth perpendicularly upon A C, and itis the
half of the Chord FH. ;

- This whcle Sine of go gr. is hereafter called Radiesy. but the other

Sines cake their Denomination frem the Degrees and Minutes of their

Arks.

Sinus verfis, the Verfed Sine is a Segment of the Diameter , inter~

cepted between the right Sine of she fame Ark, and the Circum-
’ ference of the Circle. So DC is the Verfed Sine of the Ark CB,
and GF the Verfed Sine of the Ark BF, and GH the Verfed Sine of
the Ark BH. :

A Tangent is aright Line perpendicular to the Diameter drawn by the
one Extreme of the given Ark, and terminated. by the Secant drawn from
the Center, through the other Extreme of the faid Ark.

A Secant isa right Line drawn from the Center, through one Extrenae

of the given Ark, illic meee with the Tangent raifed from the Diameter

at the other Extreme of the faid Ark. . ,

So if the given Ark be C E, or the given Angle be C AE, let the Di-
ameter be drawn eMfough the Center AtoC, andinC to AC, beraifed
a Perpendicular CI. Then let another Line be drawn from the Center
-A through E, till ic meet with the Perpendicular CI in I; . the Line

Cis aTangent, and AT is the Secant both of the Ark CE, and of

the Angle CAE.

CHAP.

CHAP. II. A

Of the general Ufe of Sines and Tengents,

1. The Radius being known, to find the right Sine of any eArk
or Angle. 7

ib the Radius of the Circle given be equal to the lateral Radius, that
is to the whole Line of Sines on the Seétor, there needs no farther
work, but to take the other Sines alfo out of the Side of the Sector
But if icbe either greater or leffer, chen let ic be made a parallel Ra- -
dius, by applying ic over in the Lines of Sines, between 90 and go
fo te Parallel taken from the like lateral Sines, thall be the Sine re~
quired. ' .
Asif the given Radius be A C, and it were required to aa:
g6gr. and his Complement agreeable to that Radins, pen eis

A FO AS ie BONS th HA

@ ener}

BLGRT Es ASKCUTICIZHCiasaegeys

avant woyecs GIG Hte yeane
eres
auceseaisze

ast Yu seagios

esug iy esuviEs
Qaggzesens eu

ae OSBE TUS CS) ERURFULE) Cooea sATH VST TS

ees teraspeieree

Let

The general Ufe of Sines and Tangents. 47

Ler AB, AB reprefent the Lines of Sines on the Setor, and let BB
the Diftance becween 90 and go, be equal to the given Radius AC.
Here the Lines A 40, A§0, A 90 may be called the lateral Sines of go,
$0, and 90, in regard of their place on the fide of the Sector. The
Lines between 40 and 40,between 50 and 50,between goand g0,may be
called the parallel Sines of go, 50, and go, in regard they are parallel one
to the other. The whole Sine of go gr. here {tanding for the Semidia-
meter of the Circle, may be called the Radius. And therefore if AC
be put over inthe Line of Sines in 90 and 90, and {o made.a Parallel
Radius, his parallel Sine between 50 and go fhallbe BD, the Sine of 50

required. And becaufe so taken out of 90, the Complement is 40, his.

parallel Sines between 4o and 40 fhall be BG, the Sine of the Comple-
ment which was required.

2. The right Sine of any Ark being given to find the Radins.

Urn the Sine given into a parallel Sine, and his parallel Radius thall

be the Radius required.
Asif B D were thegiven Sine of so gr. and it were required co find

the Radius, lec B D be madea parallel Sine of so gr. by applying it over
in the Lines of Sines becween 50 and 50, fohis parallel Radius becween. ©

90 and go fhallbe A C, the Radius required.

3, The Radius of a Circle, or the right Sine of aay Ark, being. given,
and a freight Line refembling a Sine, to find the quantity of that.

unknown Sine.

- Er the Radius or right Sine given be turned into his Parallel, then.
| take the sight Line given,° and carry it parallel co the former, tll ic.
ftay in like Sines, fo the number of Degrees and minutes where it ftayech, .

fhall give the quantity of the Sine required.
As if B Dwere the givenSine of sogr. and BG the ftreight Line

iven
cata this Angle, I carry the Line BG parallel, and find ic to ftay

in no other but 49 and 4o, and therefore 40 gr. is his Quantity,

required.

4. The:

firtt 1 make B D:a parallel Sine of sogr. then keeping the.

el ee ee Ae Ae ee eo f ¢ Re Odes PS ati GRRE LM RG ANY co has Pei tL carte eased a oo) Ue aay ce NS eam des

tae Phe general Ufe of Sines and Tangents.

~~»

X

4. The Radius or any vight Sine being given, to find the Verfed Sine
of any Arkw | / Og ay: Ao yeaa :

[F the Ark, whofe Verfed Sine is required, | be fefs than he Quadrant,

take the Sine of the Complement out of the Radius, and the Remain-
der fhall be the Sinus Verfus, the Verfed Sine of chat Ark.
~ Asif A Boeing the lateral Radius, it were required to find the Verfed
Sine of 40 gr. here the Sine of the Complement is A 50, and therefore
B 50 isthe Verfed Sine required. Orif Ireckon from B at the end of
the Se&tor, coward the Center, the Diftance from go to So is the Verfed
Sine of 10 gr. from 90to 70, the Verfed Sine of 20 gr. from 90 to 60

. is the Ver(ed Sine of 30 gr. and fo in the reft. ;

If A D be thegiven Sine of 50 gr. and it be required to find the Ver-
fed Sine of so gr. here becaufe A D is unequal tothe lateral Sine of 50 gr.

~ [make ica Parallel. And firftI find the Radius A C, then the Sine of

the Complement A 40, which being taken out of A C, leaveth C 40,

for the Verfed Sine of 50 gr. which wasrequired.

But if the Ark whofe Verfed Sine is required, be greater than the Qua-
drant, his Verfed Sine alfo is greacer chan the Radius, by the right Line
of his excefs above 90 gr. . | 3

Asif AC being the Radius given, it were required to find the Verfed »
Sine of 130 gr. here theexcefs above go gr. is 40gr. and therefore the
Verfed Sine required is equal to the Radius A C and A 40, both being

fet togecher. |

5. The Diameser or Radins being given, to find the Chords of every
e/frk, |

| a Sines may be ficted many ways to ferve for Chords, 1. A Sine

being the half of the double Ark, if the Sine be doubled ; it givech
the Chord of the double Ark, a Siné of 10 gr. doubled giveth a Chord
of 20gr, and a Sine of 25 gr. being doubled giveth a Chord of 50 gr,
and fo inthereft; As here B D, che Sine of BC, an Ark of 40 gr.

- being doubled, giveth BE the Chord of BCE, which isan Ark of
S8ogr. Wherefore if the Radius of the Circle given be equal to the
Jareral Radius, lec the Se&tor be opened near unto his length, fo that both
the Lines of Sinesmay make but one direét Line: fo the Diftance on the

Sines between 10 and to fhall be a Chord of 20, the Diftance between
20and |

aii

eae aC Pea f Riek ae a ae ; Fite cc
‘ \ . f. ti 7 ke
Pe eat eee

a ¥ “Ve oe Sige > Tr 5S (A) SRA Y, ts aE SY ph RO ee a ae a ow IRA SS pe Pee eo ae Ry co!
De RON RGR EEK: FEAON PARE RUE YMG. Bor De GRENIER RR EET Ce DEC ee SEE Cee ete hei ee aoe es
os aS eae aNe Sake Tie Y. u A se Pity aon r
é ‘ , a 2 x
“ = +

-
~

The general Ufe of Sines and Tangentss ~ ig
20 and 20 hall bea Chord of 40, and the Diftance between 30 and 305
_ fhall be a Chord of 69, and fo in the reft,

2. BecaufeaSine is the half of the Chord of the double Ark, the Pro-
portion holdeth. | eae : a

te »
fe) : *» 2
of “ey a
ae a
a % :
« fs a
a °
= CR > "
= ¢ a
6 KN S FE
., ; @
% a
iz %, *
ss = a, a
Y = ‘} ‘ a
ie i % a
5 s =
2 : =
® a ¢, a
* a . *, “a
J ® s e =
5 Sad 4 ¢, - «
@ & °é ot
« = o. =
7 . : 4 % s
2 A . =
aD . > Pe, j e
epg 'G : Li "y
hd = 5 s ee

SP 60 *0 49 Fo Qo 10 3 140 QO JO 40 $0 60 $0.

As the Diameter F H untorhe Radius A H, fo che Chord BE unto
-the Sine D E, or the Chord G L untothe Sine A L, and then if the Ra- -
dius A Hbe put for the Diameter, whichis a Chord of 180 gr. the Sine
D Eor AL, fhall ferve fora Chord of 80 gr. and che Semiradius which
is he Sine-of 3ogr. thall ferve fora Chord of 60 gr. and go for the Se-__
midiameter of a Circle, and fo inthe reft. So that by thefe means we -
thall notneed:to double the Lines cf Sines as before, but only to double
the Numbers. And to this purpofe [have fubdivided each degree of the
Sines into two, that fothey might fhew how far thehalf degrees do reach
in the Sines, and yet ftand for whole degrees when they are ufed as

Chords. |

Wherefore if the Radius of the Circle given be equal to the lateral
Semiradius (the Sine of 30 gr. and Chord‘of 60 gr.) there needs no far-
ther work, then co take che Sine of 10 gr. for a Chord of 20 @r, and a
Sime of 15 gr. fora Chord of 30 gr. ee. 23

But if che Radius of the Circle given be either greater or leffer than
the lateral Semiradius, take the Diameter of it, and make it a Parallel
Chord)of 180¢7, by applying ic over the Lines of Sines between 90
Biel 4 | H and -

f Ee Seg

go § The general Ufe of Sines and Tangentse
and 90, or take che Radius or Semidiamerer, which 1s equal to:
~ the Chord of 60 gr. and make ita parallel Radius of 60gr. by-ap-
plying it over in the Sines of 30 and 30, and keep the Seétor at this.
Angle. The Parallels taken from the laceral Chords thalf be the Chords.
required, :
As if the Diameter of a Circle given were the Eine A B, and it.
were required to find the Chord of 8ogr. Firft I make AB a pa-
rallel Chord of 180 gr. or the half of it a parallel Chord of 99 ¢r. fo-
his Parallel LG, doth give me F G the Chord of 80gr. which was.
required. 3
a Seeing that as the Sine of the Complement of the half Ark is uno the:
Radius, fo the Sine of the fame whole Ark is unto the Chord of ic: If
we feck but for one fingle Chord, we may find ir without either doub-
ling the Sines, or doubling the Number. For applying over the Radius.
_ given in the Sine of the Complement of half the Ark required, his Pa-.
_ rallel Sine {hall be the Chord required. :
As if che Semidiameter of the Circle given were A C, and it were:
required to find the Chord of 40 ¢r. the half of 40 gr. is 20 gr. the
Complement of 20 g7. is 7o gr. Wherefore I make AC a parallel Sine
of 7ogr. and his parallel Sine GL, doth give me FG, che Chord of.
40 gr. agreeable ro che Semidiameter A C. 7

Having tworight Lines refemsbling the Chord. and Verfed Sine, to find:
the Diameter and. Radius. |

- Let the two right Lines given be A B, re-

~~ fembling the Chord, G Dithe Verfed Sine of
a Circle, whofe Arch A G Bis unknown, and

and lec it be required to find the Diamecer
Having two Lines given, the firft G.D,

the fecond A.D, the half of AB, we may
find a third in continual Proportion (by the

~ fixth or nineth Propofition of the Lines) and

that fhall be the Line DF (18) the Sum whereof and of GD
~ ‘Diameter GF (20) and the half thereof is the Radius (EG).

gives the

6.. The

ROP suk Sar Ve os A x

*

The general Ufe of Sines and Tangents, *y

6. The Chird of any Ark being given, to find the Diameter and Radins.

Ture che Chord given unto a parallel Chord, and his parallel Semi- -
radius fhall be the Semidiamerer, and the parallel Radius fhall be

the Diameter.

Asif F Gbe the Chord of 80 gr. I put this over in G and L, the Sine ‘
of go,.and Chord of 8ogr, and the parallel Chord of 180 gr. giveth

me A B che Diameter required.

30 40 50 B 93° 9°

y e su
@ 46 HA DANAEAAREASA

Pev eee EET

e
eaugmamedseaas

Or if I curn the Chord given into a parallel Sine of the fame quantity,
‘his parallel Sine. of the Complement of half the Ark, doth give me the
Semidiameter.

Asif FG be the given Chord of 4o gr. I put ie over in Gand L,
the Sines of ge gr. then becaufe the half of go gr. is 20 gr. and the
~ Complement of 20gr. is 7ogr. I take out the parallel Sine of 7o gr.

and it giveth me AB for che Semidiameter, agreable co that Chord
of 4ogr. |

H 2 Having

e

Shey
te

es 7 oe pee if NS * ‘ Pek s te NAPE nt tere one ae) Ceo eS. in ae CUS yh wie . “4S AVL Wee. Se Bae) eye
SRL gape tap re ent Pee gen Sk Acie AOR RG (gee Wey a pa CNR URE Re RE GRE Zien: ar RAS FF OC ET

-

Be «The general Ufe of Sines and Tangents.:
Having the Diameter of as Ellipfis to deferibe the fame upon a plain, ]

_Ifs each Semidiameter be divided, in fuch fort, as. the Line of Sines
is divided upon the Seéfor, and right Lines drawn through each divili-
on Perpendicular to thofe Semidiameters like unto Sines; The Points
where the Sines drawn through the one Semidiameter do meet the
Sines of the Complement drawn through the other Semidiameter;fhall

. be the Points through which the Ellip(is is to be drawn. |
ee Let the Diameters be A B,D E, one croffing the middle of che other a
“ inthe Point C, Divide firft-the Semidiameters CA, CB; then the a

Semidiameters C D, C E, like unto the Lines of Sines upon the Seézor,
by the eighth Propofition of Lines: Soy the Ellipfis hall be drawn
- through the Points. at the meeting of the Sines of ro and 80, of 20 and: _

. 79, of 39 and 60, ec.

a hry 8 S| ee
Seas

ae, B:
| ie
b | Or (withontthe help of che Line of Sines )we may draw the Circle
— AF-Buponthe CenterC, and Semidiameter A C, for fo, croffing the
i ass Diameter AB with feveral Perpendicular Lines continued unto ‘the
aia Circumference of the Circle, if we divide. thefe Perpendiculars on. '
oe | either: |

Gd Sia. ale ati tee Ae ame a ha a Al a ine SAL Mn et a a
, fe ee I Cet 3 ; SE Pa

The general fe of SinesandTangests. ° 93

either fide of the Diameter in fuchi fort as the greater Semidiameter

C F is divided by theleffer, inthe Point D, and draw a Line winding
through all chofe Points,the Line fo drawn fhall be che Eilipfis.»

Or (without the help of rhe Seftor ) wemay with the Radius AC,
upon the Centers:D: and E, defcribe two occult Arches meeting in
the Points K and L. Thentaking between Cand K, any Number of
of Points M Njwe may from the Centers K and L, with the Semidia-
meter M B-defcribe four occult Arches; and with the Radius A My_.
and the fame Centers K and L, ‘crofs thenvagain with other four’Ar-
ches in the Points at O.In like manner,from the fame Cencers Kand Ly
with the Radius N Bs; we may defcribe other four occult Arches; and

-. with the Radius AN, andthe former Centers crofs them again, with

four Arches in the Points at P, and fo draw che Elliplis through the
Points O:P; ec. 1 acy: 4 a
This is ( ineffeét ).as we fhould tye a thread about A and L,and
then draw iceafily fromthe Point A round abour the two former.Cen-
ters K and L, until ic were brought tothe Point 4 again: which is alfo
an ealie way to defcribe an Bllipiis. ,
The diftance of thefe former Points from either Semidiameter may
be fee down in Numbers. For fuppofing the lefler SemidtamererC D,to
be x0, the greater (CB) to be 16, Cor otherwife. divided into any

Number of known Points,) NM we have the proportion: between CG:
_ and C B, we may find che length of the Perpendicular G I. |

If the Proportion be as to 2, the Perpendicular will be 8, 66.
If the Proportion be as a.to 3, the Perpendicular will: be about:

Js 45 ,
As the greater Semidiameter C B-
cothe:-part given | CG
So 100008, the Radius CB
co the Sine of CG.
whofe Complement is. GH-
Asthe Radius i CF
to the Sine ofthe Complement GH .. _ }
.» So the Jeffer Semidiameter: CD: bin’
ro the Perpendiculars. GI

The fame may alfo be found without knowing the Sines. For the
Perpendicular GH is a mean Proportional between A Gand GBs:
which being known er) a

As C Funto,ED, fo isG H.unto.G Ip:
| 710

54 The general Ufe of Sines and Tangentis

4. Toopen the Seftor to the quantity of any Angle given.
8. The Sector being opened, to find the quantity of the Angle.

T és one thing to open the Edges of the Se€tor to an Angle, and ano-
-& therching to open the Lines on the Se€tor to the fame Angle. For
the Lines of Lines onthe one fide, and the Lines of Sines onthe other
fide, do make an Angle of 2 gr. when the Se&or is clofe fhut, and the
Edges do make no Angle at all. So likewife the Lines of Swperficies
and the Lines of Sclidsdo makean Angle of 1o gr. which are to be
allowed to the Edges.

The Linesof Linesmay be opened toa right Angle, if the whole
Line of 100 parts be applied over in 80 and 60, —

The Line of Sines may be opened toa right Angle, if the large Se-
cant of 45 gr. be applied over in the Sines of 90 gr. or if the Sine of
_ gogr. be spplied over inthe Sines of 45 gr. or if the Sine of 45 gr.be
applied over in the Sines of 30 gr.

If it be required toopenthole Lines to any other Angle, take out
the Chord thereof, and apply it over in the Semiradiws, and thofe Lines
fhall be opened to that Angle.

As if it were required to open the Sector in the Lines of Sines toan
Angle of 40 gr. take out the Chord of 40 gr. and to.it open the Setor
in the Chord of 60 gr.fo fhall theLines of Sines be opened to the Angle
required. Orif the fame Chord of 40 gr. be applied over between
50, and §0, inthe Line of Lines, they fhall alfo be opened to the fame
Angle. If it beapplied over in 25 of the Lines of Saperficies, or 125
inthe Lines of Solids, they alfo fhall be opened to the fame Angle:
becaufe the Chord of 6ogr. or Sine of ¥e gr. and 50 inthe Lines of
Lines, and 25 inthe Lines of Saperficies, and 125 in the Solids, areail
of the fame length withthe Semiradius.

Or if the Sersiradius by applied over between the Sine of 30 gr. and
the Sine of the Complement of the Angle required, ic will open the
_ Lines of Sixes tothat Angle.

Asif the Semiradius be applied over in the Sines of
Sineof sogr. it thall open the Lines of Sizes toan An

On the contrary, if the Seéor be opened to an
required to know the quantity thereof, open the
Semiradins, and fetting one foot in the Sine of 30¢

. r. turn the other to-
ward the other Line of Sines, and it fhall fall there inthe Complement

vet

gle of 40 gr,
Angle, and itbe

30 gr. andthe :

-Compaffes to the

|
|

EMEC EA NL 3) DE ORD Peete aed RSET ot at INC sl RAPP Pe INS RA Is es eee Aree BIN ye Pe Ty PRN Pie (as hee Caw
Pte oR aera Bh CoE
‘ia t ig yrs 5 ‘ 4 y >
|

The gentral Ufe of Sines and Tangexts, ei ¢5

of the Angle; if it fallon 50 gr. the Angle is 4o gr. if on 60 grsthe
Angle is 30 gr. &c. ys

Or take overthe parallel Chord of 60 gr. and meafure it in thela-
teral Chord). and ic fhall there thew the quantity of the Angle, Asif
the Seéter being opened to an Angle, I fhould take over the Parallel
of 30 gr. of the Sines, and 6o gr. of the Chords, and meafure it in the.
lateral Chords, find itto be 40 gv. the Angle comprehended between
the Lines of Sinesis 40 gr. bus che Angle between the Edges of the:
Sector ts 2 gr. lefs, and therefore but 38 gr.

9. To find the quantity of any Angle given.

Lr out of the Angular Point, to the quantity of the Semiradiu, be
defcribed an occult Ark that may. cut both fides of the Angle, the:
Chord of this Ark meafured inthe lateral Chord, fhall give the quan-
tity of the Angle. -

Letthe Angle givenbe BAC: firft I take the Semiradins with the
Compaffes, and fetting one foot in A, I cutthe fides of the Angle in:
Band C; then] take the Chord BCC, and meafure it in the lateral
Chord,and I find ito be 21. gr.and 15 min. and {uch is. the quantity of
he Angie given... 3

Or if the Ark be defcribed out of the Angular Point at any other
diftance,. let the Semidiameter be turned into a parallel Chord of

- Gogr. then take the Chord of this: Ark, and carry it Parallel, till ic

crots in like Chords s. fo the place where it ftayech fhall give the quane-

rity of the Angle. s
As inthe former example, if Imake the Semidiameter A-B a paral-

_ Jel Chord of 60 gr.and then keeping the Seéfor at that Angle,carry the

Chord BC paralle!,till icftay fnlike Chords; I fhall find itto {tay in no:

other but rr gr. 15 min, and {uch isthe Angle B.A Cy.

an ra
Smee e
‘
if |
ree) A

pa. The general Ufeof Sines an

‘

/
‘
is

eee eee ine negeee as
i $ % ab fy a
\
a p Save

5 ~
&
”

pa ee

Tangents,

10, Upon aright Line, anda Point Given init, to msake om Angle equal

‘to any Angle given. |

Beans che. Poine given defcribe an Ark; cutting the fame

Line s,chen by the § Prop. afore,find the Chord of the Angle given’

agreeable to the Semidiamerer,and infcribe it into this Ark? fo a riche:
Line drawn through the Point given, and the end of this Chord, thald
be the fide that makes up the Angle. “9, :

Let the right Line given be A B, and the Point givenin ic be 4, and
lecthe Angle given be 11¢7. 15 min. Herel opentheCompaffes co
any Semidiameter AB, ( but asofe as I may conveniently to the late.
ralSemiradius ) and fetting one foorin 4, I defcribe an occult Ark
BC; then feek outthe Chord of 11 gr. 15 min, and taking it with
the Compaffes, I fer one foot in B, rhe ether crofleth the Ark in
C, bywhichI draw the Line A C, and it makes up the Angle re-
Qi dears” so. sotiyen? laa THD < avis!

11. Todivide the Circtmference of aCircle into any parts required

lt 360, the meafure of the whole Circumference, be divided by the
Number of parts required, the Quotient giveth che Chord, which
being found will divide the Circumference.

SoaChord of 120 gr. will divide the Circumference into three
equal parts ; a Chord ot 90 gr. into four parts 3 a Chord of 72 gr. inte
five parts; a Chord of 60 gr. into fix parts ; a Chord of § rgr.26 min.
_into feven parts; a Chord of 45 gr, into eight parts; a Chord of 40 gr.

into nine parts; aChord of 36 gr. into ten parts; a Chord of 32 or.
44 min, into eleven parts ; a Chord of 30¢r. intotwelve Dalle. 33

In like manner if it be required to divide the Circumference of the
~ Circle whofe Semidiameter is A B, into 323 firit Ttake the Semidia-
_ meter A B, and make it a parallel Chord of 60 gr,

3 then becanle. 360 gr.
being divided by 32 che Quotient will be 1 a |

rallel Chord of 11 gr.1s min.
into 32.

Bic here the parts being many, it were better to divide it fir(t into
fewer, and after to come over it again. As firftto divide the Circum-
ference into 4, and then each 4 parts into 8, or otherwife, ‘as the Parts
may be divided, | . ij Te aT |

ataiae 3 5 min, Y find the pax
and this will divide che Circumference

/

The general Ufe of Sines and Tangents. $7

12, To divide aright Line by extreme and mean proportion.

alone Line to be divided by extreme and mean proportion, hath the
fame proportion to his greater Segment, as in Figures infcribed
inthe fame Circle, che fide of an Hexvagona figure of fix Angles, hath
to afide of a Decagona figure of cen Angles: but the fide of a Hexa-
gon is aChord of 60 gr. and the fide of a Decagon is aChord of 369.

Lec A Bbeche Line to be divided : if I make ABa parallel Chord
of 60 gr. and co this Semidiameter find A C a Chord of 36 gr.this AC

_ , thall be the greater Segment, dividing the whole Line in C, by extreme

and mean proportion, So thar,

As A B the whole, tsunto AC the greater Segment: fo A C the
greater Segment, unto C B the lefler Sepmenc.

Or let A C be the greater Segment given: if I makethis a parallel
Chord of 26 gr. she correfpondent Semidiameter (hall be che whole
Line AB, and che difference C B the lefler Segment.

A C B

Or lee C Bbe the leffer Segment given: if Imake thisa paralled
Chord of 36 gr. the correfpondent Semidiameter fhall be the greater
Segment A C, which added to C B, gives the whole Line A B. c

To avoid doubling of Lines or Numbers,you may pue over the whole
Line inthe Sines of 72 gr. and che parallel Sine of 36 gr. thall be the
greater Segmenr, ies

Or if you put over the whole Line in the Sines of 54 gr.che parallel
Sineof 30 gr. fhall be the preater Segment, andthe parallel Sine of

18 gr. fhall be the lefler Segment,

J CHAP,

Sea : bee
Swe = - t t
s Bare “ Tes
)Saewe ‘

i

“SH NEA Sa a ay
4. GAD A AE
Pay wel paatge'®

¢

on of the Sun, and reprefeat che Parallels of Latitude.

inthe like fore, and then carefully draw a Line eh
foas it makes no Angles, the Lines fo drawn

ee
ts

CHAP. II.

Of the projeétion of the Sphere in Plano.

See Ts TD,

re Projett the Sphere in Plano, by rreight Lizes.

1,” ¥ ‘He Sphere may be projefed in Plano in {treight Lines, es inthe:
? nf Analemma, if che Semidiameter of the Circles given be divided
in fuch fore asthe Line of Sines on che Settor. |

Asif che Radius of che Circle given were A E, the Circle thereon
defcribed may reprefent the Piane of the general Meridian, which
divided into four equal partsin E, P, &, $, and croffed ar right Angles
with E A and PS, che Diamerer E &, thall reprefenethe Eguatorjand
P S,the Circle of the hourof6. And itis alfothe Axisof the World,
wheria P ftands for che North Pole, and S for the South Pole. Then
may each quarter of the Meridian be divided into go degrees from the
Equator towards the Poles. In which we number 23 degrees,30 min.the
greateft declinationof cheSun from Ero s Northwards, from Ato
vp Southwards, the Line drawn from sto w fhall bethe Ecliptick,
and the Lines drawn parallel co the Equator through and vp fhall be
the Tropick.

Having thefecommon Se€tions with the Plane of che Meridian, if
we fhell divide each Semidiameter cf the Ecliptick into go degrees, in
fuch fort as the Sines are divided onthe Seftor. The firt 30 degr. from
A towards & fhail{tand for the Sign of y. The 30 degr, next follow-
ingfor y. Thereftof 1 & Sy ec. intheir order. So chat by thefe
mears we have che place of the Sun for alltimes of the year,

It again we divide A P, AS, in the like fort, and fee thereto the
Numbers 10, 20, 30, @e. unto 90 degrees, the Lines drawn

n through.
ne 4 = . 2 rt,
eacliof thefe degrees parallel to the Equator fhall thew the declinaci-

if farther, we divide AE, A &, and esch of bis Parallels equally
rough each 15 degr,
fhall be Eliptical, and

reprefen:

‘

Of the Projestion of the Sphere, 59
reprefent the Hour-circles. The: Meridian PES, the hour of 12 at
noon ; that next unto it drawn throngh 75 degrees from the Center, the

hours of tt and 3, that which is drawn through 60 degrees from the
Center, the hours of 10 and 2, &c. : | |

byt

To thefe we may add the months of the year, andthe days of each
month, placing Fanuary about F, AZarch about E, Fune about J,Faly about
K, September about E &, December about the Tropick of vp: and fo the
reftacccording to their Declination from the Equator.

‘Then having refpect unto the Latitude, we may mimber it from
I

2 E Northe-

60 | Of the Projection of the Sphere.

ENorthward unto Z, and there place the Zenith: by which, andthe
Center, the Line drawn Z AN, thal! the Vertical Circle, paffing
through che Zenith and Nadir, and thronga the Center at A, inthe
Points of Eaft and Weft, and che Line M AH croffing it at right An-
gles, thall reprefent the Horizon.

Thefetwo being divided in the {ame fort as the Ecliptick and the
Equator, the Line drawn through each degree of the Semidiameter.
AZ, parallel tothe Horizon, fhall be the Circles of Altitude, and
the Divifions in the Horizon and his parallels fhall give the Azi-
muh, ! |

Laftly, Ifthrough t8gr.in AN, be drawn a right Line LK parallel
to the Horizon, it fhall fhew the time when the day. breaketh, and the
endof che twilight.

Some Ufes of thisProjettion

Fea Example of this Proje@ion, let the place of che Sun be-the »

latt degree of wy, the Parallel pafling through this place is LD,
and therefore the Meridian Aleitude ML, and the deprelfion below
che Horizon ae midnight H D: che Semidiuraal Ark LC, the Semi-
noéturnal Ark C D; the Declination A B, the Afcentions| difference
BC, the Amplitude of Afcention AC. The difference between the
end of twilight and the day break is very {mall 5 for it feems the Pa-
rallel of the Sun doth hardly crofs the Line of ewiltghr.

If the Altitude of the Sunhe given, let a Line be drawn fromit
Parallel co the Horizon: fo it fhall crofs the Parallel of theSun, and
there fhew both the Azimuth andthe Hour of che day. As ifthe place

_of the Sun being given as-before, the Altitude in the morning were -

found to be 20 degrees, the Line F G drawn Parallel to the Horizon
through 20degrees in A-Z, would crofs the Parallel of the Sun in ©.
Wherefore F © fheweth the Azimuth, and L © the quantiry of Hours
from the Meridian. Ie feems to be about half an hour paft 6 inthe
morning, and yet more than half a Poine short of the Eatft.

The diftance of two places may be alfo fhewed by this Projection,
cheir Laticudes being known, and their difterence of Longitude. .

For: fuppofe a place in the Eaft of drabia, having 20. degrees of
North Latitude, whofe difference of Longitude from London, 1s found
by an Eclipfe co be 5 Hours, 2. Let Z be the Zenith of London, the

Parallelof Latieude for chat other place muft be L D, in which the .

difference

: Of the Projection of the Spheres — . 6
difference of Longitude is L©. Wherefore © reprefenting the fice
of that‘place, Idraw through @a Parallel tothe Horizon M H,crof-
fing the Vertical AZ near about 70 degrees from the Zenith, which
raulciplied by 20,(heweth the diftance of London, and that place to be
1400 Leagues, Or multiplied by 60, to be 4200 miles.

Sect. If,
To project the Sphere in Plano, by Cirenlar Lines.

iit Nits Sphere may be projected in Plano by Circular Lines, asin
the general Aftrolabe of Gemma Frifius, by che help of the Tan-
gent onthe fide of the SeéZor. ‘

For let the Circle given reprefent the Plane of the general Meridian
as before: leritbe divided into four parts, and croffed arright Angles
with E A the Equator, and PSthe Circle of the Heur of 6, wherein
P flands for the Narth Pole, and § for the South Pole: Ler each quar-
ter of the Meridian be divided into 90 degrees, and fo the whole into »
360, beginning from P,and fertting co the Numbers of 10,20,30, Cee.
goat &, 180 atS, 270at E, 360arP. The Semidiameters A P, A fee.
AS, AE, may be divided according co the Tangents of half their
Arks, that isa Tangent of 45 degrees, which is always 10000, equal
tothe Radius, thall give the Semidiameter of 90 degrees, a Tangent of
40. degrees 83910, fhall give 80 degrees in the Semidiamerer: a Tangent.
of. 35 degrees 70021 Shall give 70, @c, So that che Semidiameters may
be divided in fuch fore as the Tangent onthe fide of the Sector chedif- -
ference being only in their denomination.

Having divided the Circumference and the Semidiameters, we may
eafily draw the Meridians and the Parallels by the. help of the
Seltore
The Meridians are to be drawn through both the Poles PandS, .
and the degrees before graduated in the Equator. The diftance of
the Center of each Meridian from A,the Center of the Plane, is equal -
tothe Tangent of the fame Meridian, reckoned from the general Me- -
ridian P AS E, andthe Semidiameter equal co the Secant of the fame -
aegree,

Phe for example, If I fhould draw the Meridian P BS, which ts .
che centh from P 4S, the Tangent of togr: 17623, giveth me AC,
and the Secant of 10 gr, 101543, givethme SC, wherefore C hi the -

wEBIER »

te uae

"te Of the Projection of the Sphere.

--Center:of the Meridian, P BS, andCS hisSemidiameter; foA Fa
Tangent of zogr. 36397 fheweth Ftobethe Center of PDS, the

- twentieth Meridian from P A S,and AG a Tangent of 23 gr. 30 ™
43481, theweth Gto bethe Center of P &S, ec.

The Parallels are to be drawn through the degrees, in A P, AS, and

- their correfpondent degrees inthe seneral Meridian, The diftanceof
the Center of each Parallel from A the Center of the Plane, ‘is equal
co the Secant of the fame Parallei from the Pole,and the Semidiameter
equal

Of the Projection of the Spheres = ss OP
equal to the Tangent of the fame degree. As if I thould draw the
Parallel of 80 degrees, which is the tenth from the PoleS, firft I open
the Compafles unto A C the Tangent of 10 degrees 176 33, and this-gi-
veth me the Semidiamerer of this Parallel, whofe Center isa little
from $, in fach diftance as 101543 the Secants S C is longer than
10000, the RadiusS A.

The Meridians and Parallels being drawn, if we number the 23 degr.
20 min. from Eto & Northwards, from AE to vp Southward, the Line
drawn from to w fhall be the Ecliptick : wich being divided in
fuch fore as the Semidiameter AP, the Arik 30 degrees fram Ato @ fhall
ftand for the Sine of 3 the 30 degr, next following for yo; the rele
for Il, S SL, Cre. in their order. ; |

If farther we bave refpect unto the Latitude, we may number ic
from E Northward unto Z, and there plecetke Zenith, by whichand
the Center, the Line drawn Z AN, fhall reprefene the verticle Cir-
cle, and the Line M AH, croffing itatright Angles, fhali reprefene
the Horizon, and thefe divided in thefamefort-as AP, che Circles
drawnthrough each degree of the Semidiameter A Z,Paraltel co che Ho-
rizon,fhall be the Circles of Altitude: andthe Circles drawa through.
the Horizon and his Poles, fhall givethe Az muths. | |

Some Ufes of this Projections

Or Example of this Projeétion, letthe place of tive Sun be inthe.
beginning of sz, the Parallel paffing through this place is «x OL,.
and therefore the Meridian Altitude ML, and the depretlion below
the Horizonat Midnight H O, the Semidiurnal Ark L ©, the Sem'-.
no@urnal Ark O ©, the Declination A R, the Afcenfional difference
R ©, the Amplitude of the Afcenfion A ©,
Or if A be put to reprefent the Pole of the World, then thall
P ASE ftand for the Equator, and P @S vp for the Ecliptick, and
the reft which before ftood for Meridians, may now ferve for particu--
lar Horizons, accordingto their feveral Elevations. Then fuppofethe
place of the Sun given co be 24 degrees of wv, his Longicude fhall’ be
PI, his right Afcention PH, ‘his Declination HI, And if the! place .
given be 19 degrees of $1, his Long tude fhall be P K, his right Afcen-
fionPN, his Declination NK. Again, the Declination broughe to
the Horizon of che place, fhaillchere (hewthe Afcentional difference 5.
Amplitude of Afcenfion,and the like conclufions of the Globe. Bur.
intend.

6h Of the Projettion-of the Sphere.
intend not here to fhew the Ufe of the Aftrolabe, butthe Ufe of the
Seftor in Projection. » a apg
And after this manner may a Nofturnal be projefted to thew the
Hour of the Night, whereof I willfet down a Type for the ufe of
Seamen. he Ob | | .

%,
2,
S)
2
@
&
SX NN
>
«t
(/ ee Soo So
ad ee
my Nee ei =a mart
< ] ba Lt heed
~4 { i

Ie confifts, as you fee, of two parts, the one isa Plane divided
equally according to the 24 hours of the day, and each hour into quar-
ters or minutes, asthe Plane will bear: the Line from the Center to
XII, flands for the Meridian, and XII ftands for the hour of 12 a¢
midnight. The other part is a rundle for fuch ftars as are near the
North-pole, together with the twelve months, andthe daysof each
, month fitted tothe right Afcenfion of the ftars.s Thofe that have occa-
fion to fee the South-pole, may do the like for che Southern Conftel-
lations, and put them ina Rundle onthe back of this Plane, and foie
may ferve for all the World.

pra

The

Of the Projettion of the spheres 86g
The Ufe of this Nof&urnal.

The Ufe of this Noéturnal is eafie and ready. For lookup tothe
Pole, and fee what Stars are nearthe Meridian : then place che Run-
dle tothe like fituation, fothe day of the monch will fhew the hour of

the Nighi.
Sucr. III.
Another way to Projet the Sphere by Circular Lines.

35 He Sphere may be projected in Plane, by circular Lines, as in
the particular Aftrolabe of Fohn Stopblerin, by help of che Tan-
gent, as before.

For let the Circle given reprefent the Tropick of vp, let it be divi-
ded into four parts, and croffed at right Angles with AC the Equi-
noétial Colure, and MB the Solftisial Colure, and general Meridian,
che Center Preprefenting the Pole of the World. Let each quarter be
divided into 90 degrees, and fo the whole into 360, beginning from
A towards B. The Meridian P M or P B, may be divided according to
the Tangent of half his Ark. Soas the Ark from the North Pole to
the Tropick vp being 9@ degrees, and 23 degrees 30 min. that is 113 de-
grees yO min. and the half Ark §6 degrees 45 main. the Meridian fhall
be divided into 90 degrees and 23 degrees 30 min. in fuch fort as the
_ Tangent of 56 degrees 45 min. on the fide of the Seclor is divided
into degrees and half degrees, of which P & the Ark of the Equator
90 degrees from the Po'e, fhall be given by the Tangent of 45 degrees.
And P %the Ark of the Summer Tropick 66 degrees 3.0 msin. from the
Pole, fhal! begivenby the Tangent of 33 degrees 15 min. And the
Circles drawn upon the Center P through A and &,fhall be the Equa-
tor, and the Summer Tropick. :

Having the Equator and both the Tropicks,the Ecliptick V & om vp
fhall be drawn from the one Tropick to the other, through the inter-
fectionof the Equator and the Equino&tial Colure. And it may be di-
vided firft into twelve Signs after this manner:'PE the Arkof the
Pole of the Ecliptick 23 degrees 30 min. from the Pole of the World,
_fhallbe given by the Tangent of 11 degrees 45 min. The Center of
the Circle of Longitude paffing through this Pole BY: and *, thall

K be

: 66 Of the Projection of the Sphere. a
vit be found at D ( fomewhat below B) by the Tangent of 66 degrees
: : 30min. Then through D draw an occult Line parallelto AC, and
divide it on each fide from D, in fuch fort asthe Tangent is divided
onthe fide of the SeGtor, allowing 45 degrees to be equall co D E,

fo the thirtiech degree from D toward the right hand, fhal! berthe
Center of the Circle of Longitude paffing through E © and m, The

Kuma rte

Gi | 9°3

fixtieth degree the Center of ZEB #. The thirtieth gr. from D towards
the left hand,the Center of * E ®. The fixtieth,the Center of = ER,
And the other intermediate degrees fhall be the Centers to divide each
fign into 30 gr i.

~ #

04 ds

Of the Projection of the Spheres 67

“If farther we have refpect unto the Laticude, we may (che Meridi-
an being before divided } number it from P Northward unto.H, and .
there place the North Inter{e@ion of the Meridian and Horizcns —
chen the Complement of the Latitude being numbred from P South-
ward unto Z, fhall there givethe Zenith; and 90 degr. from Z South=
ward untoF, fhall there give che South interfeétion of the Meridian
and Horizon. The middie between F and H fhall be Gehe Center of
the Horizon Y H = F, paffing through the beginning of Y and ™, un-
lefs there be fome former errrour. | |

All Parallels to che Horizon may be found in like fore by their In-
terfetions with the Meridian, and the middle between thofe Inter-
feétions is always the Center. |

The Azimuths may be drawn as the Circlesof Longitude were be-
fore. For the Circle of che firft Verticle V Z &, will befound ac I
(fomewhat near unto B) by the Tangent of che Latitude, And if
through I we draw an occuls Line parallel to AC, and divide te on
each lide from I, infuch fort asthe Tangent is divided onthe fide of
the Seéfor, allowing 45 degrees to be equal to IZ, thefe Divifions —
fhall be che Centers, and che diftance from thefe Divifions unto Z,
thall be the Semidiamecers whereon to defcribe the reft of the
Azimuths.

Some Ufes of thw Projection.

T'Or example of this Projection, let © the place of th€ Sun given
be 10 degr. of & : aright Line drawn from P throvgh this place
unto the Equator, fhall there fhew his right Afcenfion V K, and his
Declination K ©. Then may we onthe Center P and Semidiameter
© P draw an occult Parallel of Declination, croffing the Horizon in
L, M. the Meridian inG and N. Sothe right Lines P Land P M pro-
duced, hall fhew the time of the Suns rifling and fetting, Y Q the
difference of Afcenfion, R the difference of Defcenfion, YL che
Amplitude of rifing, and = M the Amplitude of his fecting, L NM
fheweth the length of the night, Z G fhewerh his diftance from the
Zenith at noon, H N his depreffion below the Horizon at midnight.
And then having the Altitude of che Sun at any time of the day, the In-
terfection of che parallel of Aleicude with the parallel of Declination,
fheweth the Azimuth, and aright Line drawn from P through this In-
rerfection, giveth the hour of the day. sa |
: H 2 SB Cy

68 Of the Projection of the Sphereo
| Sect. IV. |
ef third way to Projeét the Sphere ia Plano, by € ircular Lines.

4. He Sphere tay be Proje&ted in Plano by Circular Lines, after.
al Wi manner of the old concave Hemifphere, by the help of the.
Tangent on the lide of the Seéfor.

For let che Circle given reprefent ehe Plane of the Horizon, let ic
he divided into four parts, and croffed at right Angles with S N the
Meridian, and E V che Verticle; foasS may ftand for cheSouth, AV
‘forthe North, E forthe Eaft, Wthe Welt-partof the Horizon, and.
the Center Z reprefent alfo the Zenith, Let each quarter of the Ho-
rizon be divided into 90 degrees, and fo the whole into 360 degrees,.
beginning from N, and fetting co the numbers of 10,20,30,¢c. 90 at
E, 180atS, 270 at W, 260atN.

The Semidiameters Z N, ZS, may be divided according to the Tan-.
gent of half their Arks: foasthe Ark from the Zegith to che Horizon
being 90 gr.and che half Ark 45 gr. the Semidiamecers are to be divi-
deed in fuch fort as the Tangent of 45 gr. aswas fhewed before inthe
fecond Proje&tion. And if from Zwe draw Circles through each of
thefe Divifions, they fhall be Parallels of Alcitude. ;

Then having reipeét unto.the Altitude, we may (the Meridian be-
ing before divided ) number it from Z to A, and there place the In-
cerfection of the Meridian and Equator. The Complement of che Lae
titude from Z unto P; fhallthere give.che Pole of she World, and 90
further from P, (hall there give the other inrerfe€tion of the Meridian
and Equator.

The middle between thefe inrerfeStions thall be A the Center of the
Equator, pafling throvgh E and W, unlefs there be fome formerer-
rour. Fhe interfections of the Tropicks depend on the Equator. From
A 23 degrees 39 min. farther fhall be ¥ che interfeion of the Meri-
dian and the Southern Tropick. From & 23 degrees 30 min. nearer
fhall be %, che Inrerfection of che Meridian and the Northern
Tropick. The Interfeftions of the other. intermediate Parallels, —
fhall be given in| ke fort, by their degrees of diftance from,the Equa-
eor,and che middle betweenthofe Inrerfeétions.is always. the Center.

The Hour Circles may be here drawn asthe Azimuths in the. third
Projection. For the Center of EP W, the. hour of @, will be found

ag.

|

of sheeProjettion of the Spheres $9

at B, ( fomewhat near unto N.) bythe Tangent of the Latitude, And
_if through B we draw an occult Line parallel unto E W, end divide

it oneach fide fromB, in fuch fore as the Tangent is divided onthe
fide of the Seétor, allowing 45 degrees to beequalto BP, and 1§ de-
grees, for every hour, thofe Divilions fhall be the Centers, and the di-
itance from the Divifions unto P, fhall be che Semidiamerers, whereon
co.defcribe the reft of che hour Circles,

_
$50 oto ‘ries

Per 7 / A
o.. he Ve :

"ens f
LI a I
oz! etna care 'o Lt

The-

Oey ; Of the Projettion of the sphere. be

ane

_ The Ecliptick may be drawn as the Equator, For the Center of
that half which hath Soushern Declination, fhall be given by the Tan.
gent of the Altitude, which che Sun hath in his entrance into W. And
the Center of the other half by. the Tangent of his Alticude, ae -
his entrance into S, and it may be divided, as inthe former Proje@ti- |
on, or elfe by Tables calculated to that purpofe. -

To thefe Circles thus drawn, if we mal add the months of the year,
and tie days of each month, as we may well do, at the Horizon, on
either fide, between the Tropicks ; this Projeétion thall be Greed for
the moft ufeful Conclufions of the Globe, as by examples following
may appear. ~

Some Ufes of thes Projection.

Re: the day of the month being given,the Parallel chat fhooteth on
A it,doth fhew what declination theSun hath at that time of the year.

- And where this Parallel croffeth the Ecliptick, there is the place of

the Sun. Or che place of the Sun being firft given, the Parallel which

-croffeth it, fhall atthe Horizon fhew the day of the month. Either

of thefe chen being given, or only the parallel of Declination, we may
follow it, firft uncothe Horizon, there the diftance of the end of che
parallel from E or W, theweth the Amplitude; the fame among the
hour-circles {heweth the time when the Sun rifeth or fetteth. Then

~ having the Alticude of the Sunat any time of the day, the Interfeétion

of the Parallel of Declination with the Parallel of Altitude fheweth

_ the hour of the days and a right Line drawn from Z, through this

UnterfeGtionto the Horizon, giveth the Azimuth.
Thus in either of thefe Projeétions, that which is otherwife moft

troublefome, is eafily done by che help of the Tangent Line,and what

Ihave faid of this Line, the fame may be wrought by Scale and Num-
bers out of the Table of Tangents. | .

re Note, that if unto any of thefe three laff Projettions, there le added an

Index equal tothe Semidiameter of the Circle, to move upon the Center
of the Projection, and the fiducial edge thereof divided according to
the Tangents of half Arks, the Semidiameters need not be divided, and

the Inftruments will then be fitly accommodated to perform many Con-
clufions of the Sphere. “e2

By |

Of the Projection of the spheres | Brae ce eee.

s72-By the formet ways of Projecting of the Sphere, the whole Are
of Dialling may be performed upon any of them, but efpecially upon
chis laft, which may be fitted to the Horizon of any place, the manner
whereof inthis place I fhallbriefly deliver. |

1. Foran Horizontel Dial.

If ftreight Lines be drawn from the Center of the Projection
chrough the incerfections of the hour Circles wich the outermoft Cir-
cle or Horizon, thofe Lines fo drawn fhall be the erue hour Lines of

an Horizontal Dial in that Letitude for which the Projeftion was

made, for the hour Circles cut the Hor:zon at thofe degrees of
diftance. 3
2. Foran Eve&h dire&t North or Soneh Dial.

If an Index be divided as the Semidiameter of the Projeétion ZW
is, on both fides, and laid upon che fame Diameter WE’, the hour
Lines of the Projection will cutthe (ame Index in fuch degrees from
the Meridisn on either fide, asthe hour Lines on fuch a North or South

Dial ought to have upon the Plane: As,”

deg. min.
28

9 19: $4 |
_ The hour : do cutthe Fromthe
Pte 30 54 Meridian,

I

2

: 3
Lines of 4 4 Index at a7 ae
7 5 66 42%.

etrue hour diftances for a North or’ South Plane
for which this Projection was

amy
‘6

And thefe areth
sn this Latitude of 51 degrees 30 mim

made.
3. Fora Vertical Declining Dial.

Suppofe an upright Plane to decline from the South Weftward 24
decr. 20 min. Such a Plane is defcribed in the third Book of this
Treatife Chap.7- If you lay your Index to 24degr. 20 min. counted

From E towards S, and there keep it fixed, the hour Lines of the
- Projection

i eS. i
Me ent
Pees

Fe Of the Projedtion of che Spheres

por:

Projection will cut the Index in thefe degrees from the Meridian, ©
either fide thereof, at which they are to be drawr upon the Dial
Plane. | | :

4. For direét Incliners. -

~ Let the Inclining Plane be projected upon the Scheme, a Ruler laid —

tothe Poleof che inclining Plane, and to che feveral Points where
the hours crofs the Plane, che Ruler will cue che outermoft Cisclein
the degrees that the hour Lines ougheto have upon fuch an inclining

‘Plane.

Thus let che Circle W 4 E, which is the Equinoétial Circle, re-
prefent a Planeinclining to the Horizon, a Ruler laid co the Pole of
che World ( which isthe Pole of the Equinoétial Circle ) and the fe-
veral interfeétions of the hour Circles wich this Circle, fhall cur the
ourermoft Circle in every fifteenth degree, and fuch diftance oughe
each hour have from other upon the Plane. |

*§. For Declining inclining Planes.

Letra Plane decline from the South Weftward 24 degrees 20 min. |

andiacline tothe Horizon Northward 36 degrees, fuch a Plane isre-
pefented in the Diagramof the feventh Chapter of thethird Book of
this Treatife, by the CircleB MD. Nowa Ruler laid uponthe Pole
of this Plane, ( which isinthe Line QH, 90 degrees diftant from M )

-and the interfetions of she hour Circles with the Plane, fhall cut the

primitive or horizontal Circle in the degrees of diftance that there-
fpeétive hour Lines of fuch e declining inclining Plane eughtto have
upon the Dial Plane. “G3
> SECT?) ¢,

Of Projefting of the Sphere upon Oblique Circles.

N the four firft Se€tions of this Chapter, Mr Gunter hath (hewed
Dhow to Project the Sphere in Plaae upon the principal Great Circles

- of the Sphere, viz. Twice upon the Plane of the Meridian, once upon

the Tropick of , or the Equinoétial, ( parallel thereto ) and laftly
upon the Horizon.
To

Of the Projection of the Spheres 3°

- Tothefe Projeétions I chink it will not be impertinent C but very.
beneficial and facisfaGtory to the Reader) to thew how che Sphere may
be Projeéted in Plano upon any Oblique Circle, as upon any Plane
whatfoever and howfoever fituare, for all or moft Diall Planes are Ob-
lique Circles, and are Horizons in fome part of the World or other.
As for Example; A Dial Plane declining from the South Weftward
a4 degrees 20 min. and inclining Northward 36 degrees (fach is the
Dial! Plane in the tenth Chapter of che third Book of this Treatife of |
Dialling) will in fome place or other bean Horizontal Plane: And by
projecting of the Circles of che Sphere in their true politions upon
this Oblique Plane, you fhall not only difcover in what Longitude
and Latitude this will be an Horizon, but will alfo delineate out unto
you the places of the Hour-lines proper for this declining inclining
Plane, in a quite different manner and formthan that which Mr Gan-
ter hath thewed how to make the Digll in the forementioned tenth
Chapter of the third Book, by drawing the Plane upon the Horizontal.
Projeétion for this Latitude. And feeing the difference of che two
ways of working are fo various, and the variety that will appear in the.
placing of the Circles of the Sphere in their true pofitions upon fuch

.an Oblique Plane cannot but be both beneficial and delightful, I fhall
here infert the manner how the fame may be efteéted, not only upon
this, but upon any other Oblique Plane whatfoever. 7
» To proceed then, Letche Circle HX OD, reprefent a Dial Plane
declining from the South Weltward 24 degrees 20 min, and inclining
Northward 36 degrees. / |

1. Draw the Diameter H O, and crofs irae right Angles with the
Line CF meeting in cheCenterQ. — |

2. Take the half Tangent of 36 degrees, the Planes inclination, and
fet it from Q to Z, fo fhall Z be the Zenith of the Place.

3. Take the half Tangent of §4 degrees, the Complement of the
Planes inclination, and fet that from Q_to B, fo thall B be the Point
‘through which the Horizon of the place mutt pafs.
~ 4, Take the Tangent of 36 degrees the Plains Inclination, and fet tt
from Q toC. Or take the Secant of 36 degrees, and fet it from
B to C, fo thall C be the Center of the Horizon H BO. ;

5. .Takethe Tangent of $4 degrees, the Complement of che Incli-

nation, and fet it from Qto F. : :
6 Take 24 degrées 20 min. out of your Line of Chords and fet shat
‘diftance from H toc, from D to EET Gus Of aye FRM
| ee

74 Of the Prejettion of the phen

9. Draw a Line chrough the Points E and W extending it till ie
_¢crofs the Line FG laft drawn, at G, fo thall G be the Center of the
Meridian of the place reprefented by P ZS.

ro. Lay a Ruler from W co Z and it will cue the Circle in«, from
which Point a fer 38 degrees 30 min, the Co-latitude to 6, and aRuler
laid from W to d will sive the Point P in the Meridian for the Pole of
the World.

Il. Set go degrees of your Chords frombto f, andfrom frog, A
Ruler laid from W tof gives © in the Meridian Circle, for the Equi-
noctial Point, and from W to g gives M for the SouthPole, anda righe
Line drawn through P Q and M thall be the Axis of the World.
42. Through the Points W 4 £ draw the Equinoétial Circle, to
find the Center whereof, |

13. Divide W E into two equal parts in R, and raife the Perpendi-
cular R T, drawing it forch till it meet with Q P being extended, here
reprefented by thetwo Lines RT and QV, whofe meeting fhall be
the Center of the Equinoétial, which QT extended would be equal
to the Secant of the height of the Pole above the Plane; Or if from
_ T you draw a Line through C it will incerfe& QV in the Center of
she Equino&ial alfo. |

14. Divide MP intwoequall parts in D, and draw DG at righe
Angles to P M, and extend D G infinitely,
15+ Upon P, atthe diftance P D( or any other ) defcribe the Semi-
“citcle L DN, and layinga Ruler from P to G the Center of the Me.
Fidian, it will curthe SemicircleL D N in L, -at which Point L begin
-€0 divide LDN into twelve equal parts, and a Ruler laid fromP
through each of thofeequal parts thall give the Tangents of 15, 30,
45, &c. upon the Tangent G D and thall be the Centers of the fe-
veral Meridians, G being the Center.of twelvea clock, or the Meri-
dian of the Place. HOA
__t5. From the Center Qshrough the Points where the feveral Meri-
dians do cut the Primitive Circle draw right Lines, and thofe Lines
fhall be the true hour Lines of a South Plane declining Weltward
a4. degrees 20 min. and inclining Northward 36 degrees in the Latitude
Of $1 degrees 30 min. ea BTN PES fit

And

jetie spheres
of the Projeltion of the s
7. A Ruler ih Z to ¢,d, and ¢ will give the Points Wy S ang

i) : Hc Rr panlielto H O,and extend A brief Synoplis of this Oblique Projec
ane F arto A ‘all ’
4 ; jl " Draw BG pet rt

and W extending ic till ie HXOD, the declining inclining NE ra or 4 an
1G bethe Center of the | Plain, | Bec, 36 doc Plaine dvough che Center
the Circle ing, from Ss Rigtton. ss sa and iS aise
4 2 ae g. d, = Co- will be the
ne Oe aay eae Ea ail
he Poiat P in the Merid’an for the Pole of NC==Tang 36d0:BCaaSe- QV andRT extende
f ¢ REGr 36 4 =P ainsinclin.and {e&ting, gives the
Chords from) to f, and from fo ec... a nine Ea rages
wy Mer ae QF=Tan.544,= Co-Plainsinc, height above the Pla
ves M for the South Pole, anda tight He—D ae-0 ¢=244,20 m= Divide MP in2 = pa
eae . ie eke pore Plains Declination, draw D G infinitely
NEE tom che eieotias Circle, to A Ruler laid from Z toc, d,e, will Upon P,at any diftance
ide W 1 . . give W, S and E in the Horizon: Semicircle L D N,
Divide W E into two equal parts in R, and raife the Perpendi- | GE-Lw Qk or || toHO. from P to G, it wil
T, drawing it ill it meet with QP being extended, here EWentended, gives Gthe Center _ in L, at L begin to
ae ove sion QV, whofe meeting thal be of the Meridian. into 12== parts.
ter of the Equinoétial, wh extended would be equal REISE laid fram W to Z. gives a. from P to thoft=p
tcant of the beight of the Pole above the Plane: Or if from ab=Chord 38 d.30 m. —=Co-Lat. the Tangents of 15
eee ain Cit will interfea& QV in the Center of | Wh gives P for the Pole of the}world upon G D,and bet
, 14. D vide MP intw a . ; | bf==fg= Chord 90 deg. the Meridians, G be
Angles toPM and IDG parts in D, and draw DG at right | A Ruler laid from W tof gives & terof 13.

; » Upon P, atthe dif Pamitely ' the Equinoét. W pe, givesMthe Linesdrawn from Q
ech ibN, ia roe wat ay other ) defcribe the Semi- South Pole, and P QM is the interfeétions of th
ridian, it will emnthe Senicick: Sa PG the Center of the Me Axis of the World. with the Primitive
to divide LDN into aia ! NinL, at which Point L begin Through W & E; draw che Equi- prefenting the declin
through each of tho » and @Ruler laid fromP noétial Circle, Plain ) thall be the
45) Oe. upon the | give the Tangents of 1g, 30, |
fall be the Centers of the fe-
veral Meridiang, ~ € the Cenrers o} e nd
dian of the Place, Oo twelve clock, or the Meri- ——<—

‘5. From the Center th : . Place this Synop/fs againft Page 74 of the Sector, fo thar ic, ¢
dians do cur the Prime Toughtbe Points where the feveral Meri- P ‘

; ; the Scheme;may lie oye when the Book is thu:
fhall be the true hoy Li Citcle draw right Lines, and thofe Lines ° yore :
94 degrees 20 my, andi (Dts Of & South Plane declining Weltward
36 agrees in the Latitude

And

, ; Oclini
OF 58 depress 30 mip, 8 Northirarg

Of the ProjeFion of she Spheti = 9
_ And from this Scheme may further be found thar, es

| deg} mle
y, The Elevationof che Meridian H A is 69 54
2. The difference of Meridians Q P & is 14 ul
g. The heightof cheScileP Ais 17 26
4. The diftance of the Subftile and MeridianAXis 4 30

Note, In like manner wlghe be inferted in this Projeltion, the Tropicks
and other Parallels of the Sans place or Declination; The Azfo
~ mssths, Alusicanthar's, the Ecliptichy and other (either fall op
great ) Ciwcles; as t inflanced inthe Scheme by the Tropich of
Cancer, which s thereon defcribed ; bat of this Oblique Projection

I bave {aid enough in the place, FE3

ta 0) Me oe ae

Ls CHAP;

- Angles, and three fides, any

AB
BD
AGF Reétangled inG, All |ECD 53 7 48 | AD 28
BE
ED

‘ C HAP. IV. zh
Of the Refolution of right-lined Trianglese

N all Triangles there be-

7 Ang. Gr. M. §. Line Parts.
ing. x parts, > viz. three... | eee

E
three. of them being given, |G. .
the reft may be foundbythe | 4 16 15 36.| FG. 28
Sectors BS Y WS : D "
_ As may appear by the |8 21 5 |
Prop. following, wherein |B 143. 7-48 | BBR 35
for our practife we.may |4PG 73 44 12 | AG 96
nfe thefe Triangles CEA, | 4CE 72 44 12 | AE 72
CEB, CED, all which |4EB 20 36 36
are ReGangled in E. And |BCE 53 7 48

the reft confilt of Oblique | BCD 106 rs 36
Angles. }ACD 126 52 12

In a Rettangled Triangle. Pik
To a) jind the Bafe, both fides being given. ae 5

_ Let the Sector be opened in the Line of Linesto.a ri
‘before was fhewed C2p.2. Prop.7. ) then take out the fi
angle, and lay them, one on one Line |
whey meet inthe Center,

ghe Angle (as:
des of the Tri- —
s the other on the other Line, fo as.

and mark how far they extend. For the Line.

taken, A

¥

Of the Refolurion of right-lined Triangles. oF

taken from the terms of their extention, fhall be the Bafe required,
wiz the fide oppolite.tothe right Angle. A Se
radd the {quares of the two fides Casin Prep. 4+ | 6
Superfic. ) and the (ide of the compound Square fhall * Note that I cal
bethe*Bafe. © : ee ie ‘ ve Longefi fide of
As if the Lines A’'E, C B, fhould be che fides about Balt ee ser
the right Angle, and it were required to find the Bafe

fabrending the right Angle.

Firft, fet the Line of Lines to a right Angle by. applying the whole

Line of ro from 6 in the one Line,to 8 in the other. Then ifthe grea-

ter of the two Lines given be lefs than the Line of Lines, [rake the
reater of them A E, and transfer it with the Compafles into one of

the Lines of Lines,and find, that, in my Seétor ( which is 14 Incheslong, »

and fo,.the Line of Lines, almoft 7 Inches) it reacheth from the Center
(0 518... . | .

Again, Itake the leffer Line C E,and transfer it into the other Line

ot Lines, and find, thatitreachech fromthe Center unto154, where-

fore Ltakethe diftance from 151 unto 518; and fuch is. the length of..

the Bafe A C required. at
If either of the Lines given be too large for the. Seffers then I may

meafure them by Feetor Inches,.as fappofe I find the length of AE
to be'about 720, and of CE 210. Then inthe Lineof Lines ( being
fet, one Perpendicular to the other, as before ) I extend the Compatfles
from 210 unto 720;and meafuring chis extent in che Line of Lines, find

itto be 750 parts, where

fore I prick down 750 parts inthe Line AC,,

i

@.

from the fame Scale by which I meafured A E, and C E. So, this Line |

AC fhall betheBafe required
In working by the Line of Superficies, I need no opening of the

Seftor. For, raking the Line CE with my Corapaffes, and meafuring |

icin the Line of Saperficies upon my Seftor, 1 find it near 13 parts.
Then taking the Line AE, I finditrtobeabout 269. Thefe wo be-

ing added cogether make 292 sand this extent is the length of the.

Bafe AC required. . | : ,
2,70 find the Bale, by having the Angles, and one of the fides givens.

Take the fide given,and tutn it into the parallel Sine of his oppofite:

Angle; forthe parallel Radius fhall be the Bafe.

As ifthe Line A E were the fide of a Reétangle Triangle-
Na oppolite-

3

“9 Of the Refolution of vight-bned Triangless
oppofice to an Angle of 73 gr. 4§, and it were required to find the
Bafes |

Firft, I take che fide A E wich my Compafies, and fet it over in the
Sines of 73 pr. 45. So, the Parallel Radius taken from between 90 and
90, will give the Bafe A Crequired. :
If the fide given be fuch as cannot well be fitted over in the Sines of
2 his oppofite Angle, I may meafure it by feet or inches, and fuppofe I
™ “find the length of AE to be 72@, then would I take 720 parts out of
the Line of Lines,and make it a Parallel Sine of 73 gr. 45. So, the Pa-
rallel Radius taken from between go and go, and meafured inthe Line
of Lines will be found to be about 750 parts: Wherefore, T prick
down 750 inthe Line AC, by the fame Scale, whereby I meafured
AE: and this Line AC fhall be the Bafe required.

3. Tofnd a fide by having she Bafe and the other fede giver,

i Let the Sector be opened in the Lines of Lines toa right Angle, and
5 ae the fide given laid on one of thofe Lines from the Center : shen take
| the Bafe witha pair of Compafies, and fetting one foot in the term of
the given fide, eurnthe other tothe other Line of she Sector, and ie
thall chere fhew the fide required. |
_ Or take the Square of the fide out of the Square of the bafe ( asin
Prop. 4. Superfic. ) and the fide of the remaining Square (hall be the
fide required. Bis i
Thus having A C for the Bafe, and CE, for the fide of a Reétangle
Triangle , the other fide will be found tobe A E,
‘Gr, if A C, being meafured, be750, and CE, 210, the other fide
A & willbe found to be 730. 4 oH :

Tofind a fide having the Bafc, and the Angles given.

m _ . Take the Bafe given, and make it #Parallel Radius, fo the parallel
Sines of the Angles, thall be the oppofite fides required. 0

Thusin the Re&angle AE C, if AC be madea Parallel Radius,the
Parallel Sine of 73 gr. 45, will give the ide A, E; and the Parallel
Sine of 16 gr..¥'. will give the other fide E,

5 Zo

~

Of the Refution of righi-lined Trisnghs) ag.

s. Tofind a fide by having the otber fide andthe Angles givens

Take the fide given, and turn it into his Parallel Sine of his oppofice
Angles; fothe Parallel Sine of the Complement fhall be the fidere«
quired. | |

Thus in the Re&angle D EC, if CE be made a Parallel Sine of
§3.er- 9m. the parallel Sine of 36 gr. 52 . will give the fide E D. and
the Parallel Sine of 90 gr. willgive the Bafe C D. |

6. To findthe Angles by having the Bafe and one of the fides given,

Firft, take out the Bafe given, and laying it on both-fides of che
Seétor, foasthey may meetinthe Center, and mark how far it ex-
cendeth. Then take outthe lateral Radius, and to it open the Sedtor
in cerms of the Bafe, This done, take out the fide given, place
it alfoon the fame Lines of the Sector fromthe Center. For the Pa-
rallel taken in che tecms of this fide, fhall be she Sine of his oppofice
Angle. .

" Ortake the bafe given,and make it a Parallel Radius; thentake the
fide given, and carry it parallel tothe Bafe, till it ftay in like Sines : fo.
they thall give the quantity of the oppofice Angle. |

Thus inthe Reétangle A E C,having the Bafe A C,and the fide A E,
you may find the Angle CA E,to be 16 ¢r. 15 m.

7. Tofind the Angles by having both the fides given,

Take out the greater fide,and lay it on both fides of the Sefer, fo.
as they meet inthe Center, and mark how far it extendeth. Thentake
the other fide, and to it open the Sector in che terms of the greater.
fide; fo the Parallel Radius thall bethe Tangent of che lefler Angle..
The third Angleis always known bythe Complement;, .

Thus inthe Re&tangle D EC, having the fidesC E, and ED, you
_ may find the leffer Angle ECD tobe 36 gr. 52. and thereforethe
other AngleEDCrobe 53.4.8 m,

8, The eats being given, to find the Tangent and Secant of any

9, The

.

"35 Of the Relolation of right-lined Triahgles:

a

he > a

9. The Tangent of any Ark being given, to find the Secant thereof, and
the Radius. A je ep eed Cae
10, The Secant of any Ark being given, to find the Tangent thereof,
and the Radius. —

The Tangent, and the Secant, together with the Radius of every
Ark, domskearight Angle Triangle; whofe fides are the Radius and

Tangent, and the Bafe always the Secant; and the Angles always

known by reafon of the given Arks, Asinthe Rectangle AEC, ifon
the Center A, and Semidiameter AE, you defcribe aCircle, then
make-A-E, tobe the Radius, and EC, aTangent of 16.15 and A Ca

Secant of 16 gr. 15 2.

If you defcribea Circleon the Center-C, aad Semidiameter C E,
then is C Ethe Radiusand E A, a Tangent of 73 gr.45-m. andC Aa

Secant of 73.45.

Wherefore the Solution isthe fame with thofe before.

In any right-lined Triangle whatfoever, ,
11. To find afide by knowing the other twe fides, and the Angle con
tained by them. \e

Let the Seé&or be opened in the Lines of Lines tothe Angle given
4s | fhewed before, Cap. 2. Prop. 7. Then take ouc the fides of the Tri-
angle, and laying them the one on the one Line,the other onthe other,
fo asthey meet inthe Center,. mark how far they extend. For the
Line taken between che termsof their Extenfion, {hall be he third fide
required. | | |

Asif AC and A D were two fides of aright lined Triangle con-
taining an Angleof 16 gr. 16 m. and it were required, to find the third
fide fubtendingthis Angle. |

Firft, fer the Linestoan Angleof 16. 16m. by applying the Sine
of 8 gr. 8m. over in the Points of 50 and 50, inthe Line of Lines.
That done, Itakethe longer Line AD, and transfer it with my Asad
pafles into one of the Lines of Lines, and find ittoreach fromthe Cen
ter (0 720, ee 4 p> fades a helbee
Again, I take the leffer Line A C,and transfer it into the other Line
of Lines, where ie reacheth fromthe Center.to 540, whereforeltake
the diftance from 540to 720, and fych is thé length of she third fide
C D required: | iy" s

AO Or

‘ten oa

Of the Refolution of right-lined Triangles; = 3
Or (if the Lines be given in meafure.}) A D 100, and AC 75, Tex"
tend the Compafles from 100075, and meafuring chis extenc inthe
Line of Lines, findtobe 35. Whereupon I take 35 parts oucof the

Scale, by which AC, and AD were meafured, and prick them
down inthe LineC D. So, this Line C D fhall be the third fide

required.

12.70 find a fide by having the other two fides, and one of the adjacent An-
gles, (0 it be known which of the other Angles is Acute or Oblique.

Let the Sector be opened in the Line of Lines to the Angle given, and
the adjacent fide laid on one of thofe Lines from the Center ; then take
the other fide witha pair of Compafles, and fetting one foot in the term
of the former given Blech the other co the other Line of the Seétor
which here reprefenteth the fide required, and it fhall crofs it in two

laces; butwith whichof chem isthe term of the fiderequired, muft
be judged by the Angie. ,

Asif in the Triangle following, the fide A C being given, andthe
fideC D and the AngleC A D 16 gr.16 m. it wererequired to find the
fide A D.

Firft, I open the Se€tor in the Line of Lines to an Angle of 16 gr,
16 m. and laying the adjacent fide from the Center A, find where ig
extendeth in C. ThenI take the other fide C D with the Compaf-
fes, and fetting one foot in C, and turning the other tothe other Line
of the Seétor, I find that ie doth crofs it both in Band D. |

Or, (if the Lines be giveninmeafure ) A C75, and C D 355 I may
take 35 out of the Line of Lines, and fetting one toot in 75, I thall find
the other foot cocrofs the other Line of the Seétor, both at 44 ( an-
fwerable to AB )and at 100(anfwerableto A D.)

_So that it is uncertain whether the fide required be A Bor A D, on-
ly it may be judged by the Angle. For if the inward Angle where
they crofs be Obtufe, che fide required is the leffer ; if ic be Acutey ie

___ isthe greater. 7 |
13. Tofind a fide by having the Angles, awd one of the other fides given

Take the fide given,and turn ie into the Parallel Sine of his oppo-
fite Angle; {o the Parallel Sines of the other Angle fhall be the op-

ofire fides required.
4 : M 4 As

S2 Of ihe Refolution of right-lined Triangles,
Asif in the Triangle ABC, having tie fide A D, and knowing
the Angle C A Bio be 16 gr. 16 m. andche Angle A BCtobe 143
deg. mm, it were required to find tie ocuertwo fides, AC,and BC. .

The three Angles of a righe-lined Triangle, are always equal to
180 gr. wherefore Tadd 16 gr. 16m. unto 143 gr. 8m. and by the
remainder to 180 gr.find the third Angle A C B oppofite to the known
fide A B, tobe 20 gr. 36 m. ThenI take che fide A B, and make ita
Parallel Sine of 20 gr. 36m,

So, his Parallel Sine or 16 degr. 16 m. will be the the fide BC 3 and
the Parallel Sine of 143 degr. 8 m. willbe the fide A Ce

Or if meafuring the lide A B,I find itto be 44; I may take 44 parts,
either out of the Line of Lines, or ot of any other Scale of equal
parts, and make it a Parallel Sine of 20 gr. 36 m, So bis Parallel Sine of
16 gr, 16 m, meafured in the fame Scale, will give 35 for the length of
the fide BC: and the Parallel Sine of $6 gr.52 m. will give 755 for the
length of che other fide A C. :

When the Angle comes to be above 90 gr, the Sine of 8ogr. doth
ftand for aSine of 100 gr. and the Sine of 7ogr. for a Sine of ILO gr,
and fothe reft; for chofe, which are their Complements to 180.gr.

14. To find the proportion of the fides by having the three Angles.

Take the lateralSines of the Angles, and meafure them inthe Line

of Lines. For the numbers belonging tothofe Lines do give the pro-
portion of the fides.

- Thas; in the two equi-angle Triangles AE C,A G F, if you takethe:
Jateral Sine of gor. for the right Angle at EandG, and meafureit
inthe Line of Lines,you fhall find-it to be roo, Then take the lateral
Sine of 16 gr,16 m. for the common Angle at A, you fhall:finditto:
be 28. Take the lateral Sine of 73 gr. 44 m. For the third Angle.at C
and F, you fhall find ittobe 96. Such therefore is the proportion of
she fides. eH
As 100: 96,28:: Soare75: 72. 21.

15. To fisd.an Angle, by knowing the Three fides.
Let the two containing fides be laid on the Lines of the Sefor, from

the Center, one on one Vine, and the other on ehe other; and let the

third fide, which is oppofite cothe Angle required, be ficced. over.in

their terms, fo fhall the Seétyr be opened in thofe Lines to the quantity. |

of che Angle required.. The

es ee ee ee

-
*

| Of the Refolution of right-lined Triangdes. 83

The quantity of this Angle is found as in Cap.2. Prop 8, Thus having
the three fides of the Triansle ACD, tofind the Angle at A, Itake
the two containing fides A D,AC, and transfer them with my Com-
paffes into the Line of Lines: where I find the one to reach from the
Center, 072; theother,to 54. |

Then I take C D, (the lide oppolice tothe Angle at A) and fet that
over between 72 and 54,

Or if the three fides be given in meafure A D 103; AC 75;CD35:
I might take 35 for the fide C D ont of roe Line.of Lines, and fer thae
over from 100 to 75. This done I take the diftance between goand 56 °
and meafuring it in the Line of Sines I find ittobe about 8 gr. 8m. the
double whereof is 16 gr. 16 the Angle required.

16. To find an Angle, by having two fides, and one adjacent Angle.

Firft take out the fide oppofite to the Angle given, and laying it on
both fides of the Sector, fo as they meee in the Center, mark how far
itexcendeth ; then take out the lateral Sine of che Augle, and to it
} open the Seétor in che cerms of the firft fide: this done take out the
| other fide given, and place it alfo onthe fame Lines of che SeStor frons
_ the Center, for the Parallels taken in-che terms of this fide; thall be
_ the Sine of she Angle oppofite co the fecond fide.

Or take out the fide oppofite to the Angle given, and make it a Pac
ralleMineof that Angle: then cake the other fide given and carry it
Parallel to che former, till it ftay inlike Sines: fo they fhall give the
| quantity of the Angle oppofite to the fecond fide,

_ Thus in the Triangle A C D, knowing two fides A C, CD, with the
Angle C A D oppofice to the fide C D,you may find the Angle A DC
| Oppofite to the other known fide A €, co be about 36 gr. 52 m.

17. Ti 4 find an Angle by having two fides, and the Angle contained by

them.

Firft find the third fideby the 1 Prop, and then the Angles iflay be

found by the 15 or 16 Prop.
__ For obfervation of Angles, the Seftor may have fights fet onthe

| movable foot: fo that by looking through them, the edges of the Sector
‘may be applied to the fides of the Angle.

M 2 For

84 Of the Refolution of right- lined Triangles.

For meafuring of the fide of leffer Triangles, ~ Linke ‘
any Scale may fuffice, either of feer, orinches, or | -
leffer parts. But for greater Triingles, efpecially ©
for plotring of grounds, I hold ie fit coule achain
of four Perches in length, each Perch divided in-
to 25, and the whole Chain an hundred Links,
wherein, ifche whole Chain be (according to 16 7
foot in a Perch ) 66 foor, (that is 792 inches )
each feveral Link will be feven inches and 73 <=.

If (according to 18 in the Perch) the whole
Chain be 72 feerinlengrh (thatis, 864 inches )
- when each feveral Link will be eight inches
and $o5

‘For fo the length being multiplied into the
breadch, the five laft Figures give che content in
Roods and Perches by this Table; the other Fi-
curestowards the lefthand do fhew the number
of Acres directly. .

As in a long Square, where the length is 24
Chains 4 che breadth 13 Chains the ufual way
is, to refolve the Chains into Perches: So the
length is 97 Perches, and the breadth 54 Perches.
Thefe multiplied one into the other make 5238
fquare Perches, and thofe ( divided by 160 ) give
32 Acres, 2 Roods, and 38 Perches for the content
required.

Bur, reckoning by Chains and Links, the length
is 24 Chains 25 Links, the breadth 13 Chains 50
Links. Thefe multiplied one into the other make

CHAP:

| oe
GHAP. V.
Of the Refolution of Spherical Triangles

NOr our practife in Spherical Triangles, let A be the Equino Rial

Point, A B an Ark of the Ecliptick, reprefenting the Longitude

of the Sun inthe beginning of ¥,B Can Ark of the Suns Declination

fromthe Equator, and A Can Arkof the Equator reprefenting, the.
right Afcention. | ,

Let B D be an Ark of the Horizon, reprefenting the Amplitude of
che Suns rifing fromthe Eaft, and B Ean Ark of the Horizon for his
fetting from che Weft: foDC fhall be the difference of Afcenfion,
and € Ethe difference of Defcenfion; AD the Oblique Afcenfion,.
and A Ethe Oblique Defcenfion of the fame place of the Sun in our
Latitude at Oxford of $1 gr. 45 m. whofe Complement 38 ¢r. 15 me.
isthe Angle at E andD. The Triangles A CB, DCB, ECB, are:
Redtangled inC : theother ADB, AEB, confift every way of Ob-
lique Angles,

Or, to it an Example nearer to the Latitude of London, Let Z. PS res
prefent the Zenith, Pole, and Sun, ZP being 38¢r, 30 m. the Comple-.

. Men:

is: ;
i = bes as ;

86 Off the Refolution of Spherical Triangles.

ment of the Latitude, PS 70 gr. the Complement of the Declination,
and Z§ 40 gr. the Complement of the Suns Altitude. The Angle ae
Z fhall thew the Azimuth, and the Angle a¢ P, the Hour of the Day
from the Meridian. Then if from Z to PS we let down a Perpendicular
ZR, we thall reduce the Ooslique Triangle into rwo Reétangle Tri--
anglesZ RP, ZRS. Orif fromSto Z P we fet down a Perpendicu-

lar S M,we fhall reduce the fame Z PS into two other Triangles, § M 7
SMP, Reétangled at M: whatfoever is {aid of any of thefe Triangles.
the fame holdeth for all other Triangles in the like cafes. ‘
For the Refolution of each of thefe, there be feveral ways. Yonly
chufe thofe which are ficteft for the Seéter, wherein ifthat be remem-

bred which before is fhewed in the general ufe of the Seay concernin

Lateral and Parallel entrance, it may fuflice only to fee down the Pro.

polition of the three parts given, to the fourth tequired, and f
firft by the Sines alone. x : 0 fhew

~-

fy

Of the Refolution of Spherical Triang les, $7
Ina Refkangled Triangle.

I. Tofind a fide, byknowing the Bafe, and the Angle oppofite to the re~
quired fide.

Asthe Radius
isto the Sine of the Bafe:

So the Sine of the oppofire Angle
to the Sine of the fide required.

* Asin the Rectangle A C B,having the Bafe A B, “ln ReéfangledTvi-
the place of the Sun 30 gr. from the Equinoétial 4”g/es.the fide oppo~
Poinr,and the Angle B. A C of 23 gr. 3.0 m the grea- ab tee Be
teft Declination, if it were required to find the fide "°°" "7" 04
B C the Declination of the Sun.

Take either che lateral Sine of 20 gr. 30m. and make it a Parallel
Radius; fo the Parallel Sine of 30 gr. raken and meafured inthe fide
of the Sector, fhall give che fide required 11 gr.30m. Ortakethe
Sine of 3ogr. and make it a Parallel Radius ; fo the Parallel Sine of
23 gr. 30m, taken and meafured in the lateral Sines, fhall be r1 gr.
3.0 m, as before. | ;

So inthe Triangle Z P S, having Z.P 38 gr. 30 m, and the Angle P
3° gr. 34m. given, we fhall find the Perpendicular Z Rto be 19 Ors.
ib ms. or having P'S 70 gr. and the-faid Angle P 31.gr..34 a, given, we
may find the Perpendicular SM to be 29,gr. 28 m.

a Tofind the fide by knowing the Bafe andthe other fide.

Asthe Sine of the Complement of che fide given
istothe Radius: :

So is the Sine of the Complement of the Bafe
co-the-Sineof the Complement of the fide required.

So inthe Reé&angle AC B; having A B ZO gre. and B Ee rl gr, 30 m3.
given, the fide A C willbe found 27 gr.54 m, ike

Or inthe Re&tangleZ R P, having Z P 38 gr. 30m, andZR298r0
3m. given, the fide R P will be found 347.7.

3.7

38 | Of the Refolution ‘of Spherical Triangles,

3. To find A fide, by knowing the two Oblique Anglet,

_ Asthe Sine of either Angle

co che Sine of che Complement of the other Angle.
So is the Radius
to the sine of the Complement of the fide oppolite to the fecond
Angie. ‘

So inthe Re&tangle A C B having C AB for the firft Angle 23 ry
30m. and AC B, for the fecond 69 gr. 220m. the fide A C willbe
found 27.g7.54 m. Or making ABC che firft Angle, and CABthe

fecond, the fide B C willbe found 13 gr. 30 wm. :

4. Tofiad the Bafe, by knowing both the fides.

‘As the Radius :
to the Sine of the Complement of the one fide :
‘So the Sine of the Complement of the other fide,
“to the Sine of che Complement of che Bafe required,

‘So inthe Re&tangle A C Bhaving AC 2797. 54m. and BC x1 or:
30m. the Bafe A B will be found 30 gr. et jae II gr:

Ԥ$; To find the Bafe by knowing the one fide, and the Anghe oppofite to that

fide. |

As the Sine of the Angle given, —
tothe Sine of the fide given:

So is the Radius hah
co the Sine of the Bafe required.

Sointhe Retangle B C D, knowing the Latitude and the Declinati
on, we may find the Amplitude 5 as having BC the fide oF ihe Dealt
nation 11 gr.307m. and B DC the Angle of the Complement of the
Laticude 38 gr. 15 ~. the Bafe B D, which is the Amplitude, willbe

found to be 18 gr. 47 m.

Ge Zo

oe Fy eee er PO SN REN eM eel oy PU Seu rh ey WOR Ce ee ae SR leg Tc Sa Ren ye i See a a Rn i ore nd Me ee Arty
; he > 6 gas Pry : 4 \ 3 :
aa : ’ Sg
\ ‘ 4 ¢ pe yp
| . > ”
} .
|
| K

|
|
|

|

ae pofition; fo shat if she one will not hold, ee other may.

Of the Refolution ‘of Spherical Triangles; «8g
6. To find an Angle}, by the other Oblique Angle, and the fade oppofire tothe
ingsired Angle. ey
Asthe Radius,

tothe Sine of the Complement of the fide:
So the Sine of the Angle given, Fate a
to the Sine of the Complement of the Angle required. «©

So inthe Reétangle AC B, having the Angle BAC 23 gr2 30m,

and the fide A C 27 gr. $4. the Angle ABC willbe found 69 gr.

25 MA

Pal

7. To find an Angle, by the other Oblique Angle, and the fide oppofite tothe
Angle given. ,

As the Sineof the Complement of the fide

co the Sine of the Complement of the Angle giver :
Soisthe Radius, ™

-cothe Sine of che Angle required. -

So in the Reétangle A C B, having B A C 23 gr.30 mw.and BC ir gr.

ve 30 m.the Angle A B C willbe found 69 gr. 21 m,

8. To find an Angle, by the Bafe, and the fide oppofite to the inquired
Angle. |

As the Sine of the Bafe
is co the Radius :

So the Sine of the fide
cothe Sine of the Angle required.

So in the Regtangle BC D, having B. D18 gr. 47m. and BC 11 gr.
30m. che Angle B DC will be found 38 gr. 15 m. . | |
Thefe eight Propofitions have been wrought by the Sines alone ;
thofe which follow require joynt help of the Tangent.
And forafmuch asthe Tangent could noe well be extended beyond

63 pr. 30m. I thall fec dowa two ways for che refolution of each Pro-

9.79

eae ee ee

-—_

96 Of the Refolution of Spherical Triangles:

' 9. To find a fide, by having the other file, and the Aagle oppofite to the
ingnired fidi« . ca dhioed

1. As the Radius
tothe Sine of chefidegiven: | .

So the Tangent of che Angle, .
co the Tangent of the lide requited. in

2. Asthe Sineof the fide given,
sco the Radius:
Sothe Tangent of the Complement of the Angle
co the Tangent of che Complement of the fide requir¢d.

So inthe Rectangle A C B, having the fide AC 27 gr. $4.m.and the
Angle B A C 23 gr. 30 m. the fide BC will be found to be tf gr.30 m.

10. To find a fide, Ly having the other fide, and the Angle next the ingui=
ved fide. |

As the Tangentof the Angle,

to the Tangentof the fide given =.
So isthe Radius

to the Sine of the fide required.

a. As the Tangent of the Complement of the fide, -
to the Tangent of the Complement of the Angles
So 1s the Radius =
to the Sine of the fide required.

~ Fhis and the like, where che Tangent ftandeth in the fart place, are
beft wroughs by Parallel entrance. And fo in the Rectangle B C D
having BC the fide of Declination Il'gr. 30m, and BDC the Angle
of the Complement of the. Latitude 38 £r> 1§ m. the lide D Cy which
is the Afcentfional difference, will be found 14 gr. Sm.

By the Afcenfional difference is g'ven the time of che Sans rifing
andfetting, and length of the day : ailowing an hour for each 15 gr,
and four minutes of time for each feveral'degree. As in gr

the difference betweenthe Sans Afceafion in aright Sphere, which is

always.

the exam We; :

~ ge
yo

Of the Refolution of Spherical Triangles: 9%
always at fix of theclock, and his Afcenfion in our Latitude being
14gr. 57 m. it fheweth that che Sun rifech very near an hour before
fix, becaufe of the Northern Declination; or after fix, if the Sun be
declining to the Southward, |

1x. To find a fide by knowing the Bafe, and the Angle adjacewt wext tothe
ingnired fide. loa 7

1, Asthe Radius, — |

To che Sine of che Complement of the Angle:
So the Tangent of the Bafe,

co the Tangent of the fide required, .

2. AstheSineof the Complement of the Angle,
is cothe Radius:
Soisthe Tangentof the Complement of the Bafe,
so the Tangent of the Complementof the fide required.

So inshe Re&tangle A C B, knowing the place of the San fromthe
next Equinoftial Point, and the Angle of his greaceft Declination, we
may find his right Afcenfion : viz. the Bafe A B 30 gr. and the Angle

BAC 23 gr. 30 m.being given, the right Afcenfion A C willbefound ~

(27 gre 54 me
2, To findthe Bafe byknowing the Oblique Angler.

As the Tangent of theone Angle,
tothe Tangent of the Complement of the osher Angle :

So is the Radius,
ro the Sine of the Complement of the Bafe.

Soin the RectangleA CB, having BAC 23 gr. 30m. and ABC
6ggr. 21m. the Bale A Bwill be fonnd 30 gr.

13. To find the Bafeyby one of the fides, and the Angle adjacent next that fdee

r. As the Radius,
istothe Sine of che Complement of the Angle:
*

N 2 So

92 Of the Refolution of Spherical Triangles.
- » So the Tangent of the Complement of thefides) = ©
+» tothe Tangent of the Complement of the Bafe.

2. As the Sine of che Complement of the Angle,. »
isto the Radius: :
“ So the Tangent of che fidegiven,.
co the Tangentof the Bafe required.

So in the Reftangle AC B, having AC 27 gr. 54. and BAC:
a3 gr. 30 m. the Bafe A-B willbe found 30 gr: om,

14. To find an Angle, by knowing both the fides.

“ t. Asthe Radius,

7 is to the Sine of the fide next the inquired Angle:

oO Sothe Tangent of the Complement of the oppolite fide,
coche Tangent of the Complement of the Angle: required..

rong 2, As the Sine of the fide next the inquired Ang'e,.
isto the Radius: ,
Sothe Tangent of the oppofite fide,
tothe Tangent of the Angle required..
Soin the Retangle A C B,having A C 27 ¢r. 54 m. and BC t1gr,
com 30 m. the Angle at A will be found 23.¢r. 30m. and the Angle at B.
by Gegr. 21m.

¥5. Jo igs the Angle, by the Bafe, and the fide.adjacent tothe required:
agle.

i: t, Astbe Tangent of the Complement of the fide,.

Bee" to the Tangent of che Complement of the Bafe:

‘ $oiscthe Radivs, - 3
co the Sine of the Complementof the Angle required, a

, As the Tangent of the Bafe,
ro the Tangent of the fide:

, So ts the Radius,
a | co che fignat the Complement of she Anele required.
re ° :

Of the Refolution. of Spherical Triangls, = § = 93.
Soin the Regtangle BCD, having tke Bafe BD 16 gr. 47 m. and
the fide BC 11-gr.30 m..the Angle D.BC between them will be found
53 gre l§ wm. } ne |
46. To find an Angle, by knowing the other Oblique Angleyand the bafe.

1, As the Radius, |
tothe Sine of the Complement of the Bafe s
Sothe Tangent of the Angle given,
To the Tangent of the Complement of the Angle required..

i As the Sine of the Complement of: the Bafe,.
~ istothe Radius:

So the Tangent of the Complement of the Angle given. me
co the Tangent of the Angle required. ae
So inthe Reétangle ACB, having the Angle at A 23 gr. 30m. and —
- the Bafe A B 30gr. the Angle A B Cwill be found 69 gr. 12 m.
-Thefe fixteen Cafés areallthat: can fall oucina Rectangle Triangles:
thofe which follow do hold |
In any Spherical Triangle what{oever. bdo
17. Tofind a fide, oppofite toan eAngle given, by knowing one fide, andi
swe Angles, mbereof one % oppofite tothe given fide, the other to the:
fide required. -
Asthe Sine-of the Angle oppofireto the fide given, | Re
is co the Sine of the fide given: |
So the Sine of the Angle oppofite to the fide required). ae
tothe Sine of che fide required.. er “
So in the Triangle A BE, having the place of the Sun, the Latitude;, ‘i
and the preateft Declination, we may find the Amplitude. Ashaving. sai
AB30gr. BAE 23-gr. 30m. and AB E 38 gr. 15m, the fide BE %
which is the Amplitude, will be found 18 gr.47 m, pe
~ ab.Te 4 (one “fk

es

Pcs

"94 | ‘Of the Refolution of ‘Spherical Triangles. |

18. To find an Angle oppofite to a fide given, by having one Angle ana
two fides, the one oppofite to the given Angle, the other to the An-
gle required. | ‘ ;

As the Sine of the fide oppofite to the Angle given,

is tothe Sine of that Angie given:
So che Sine of the fide oppofiteto the Angle required,
to the Sineof the Angle required.

Sointhe Trangle Z P'S, having the Azimuth, and Latitude,and De-
-clination, we may find che hourof the day, As having P ZS 130 gr.
gim.PS Jogr. and ZS 40 gr.the Angle Z P S,which fheweth the hour
from the Meridian, fhall be found 31 ¢r. 34m, » i

19. Tofind an Angleby knowing the three odes.

This Propofition is moftufeful, bue moft difficule of all others: as
in Arithmetick, fo by the Seéfor, yer may it be performed feveral
1. According to Regiomontanus and others,

As the Sine of the leffer fide, next the Angle required,

to che difference of the verfed Sines of the Bale, and difference of —

So is the Radius, (the fides:
to a fourth proportional, Any

Thenasthe Sine of the greater fide nextthe Angle required,
is co chat fourth proportional : | -
So isthe Radius, |
_ tothe verfed Sine of the Angle required.

Soin the Triangle Z P'S,having the fide PS,:the ent 0:
the Declination 70 gr. om. ‘the fide ZP the ialeicciontuane La
cittide 38 gr. 30m. andthe Bale ZS, che Complement of che Altieude
4ogr. the Angle of the hour of the day:Z PS willbe found 31 ¢r
34 m. which is 2 4. 6 m. from the Meridian. 3

For the Bafe being 40 gr.e m, and che difference of the fides 38 gr.
30m. and 70 gr. Om, being 3% gr. 39 m, the difterence of their verfed

Sines

Of the Refolution of Spherical. Triang les. (Oe

Sines will bé the fame with the diftance between the right Sine ot
go gr.and 58 gr. 30 m. This difference I take out,and make it a Paral-
lel Sine of the fefler fide 38 gr. $0 m. fo the Parallel Radius will be the
fourth proportional. Thencoming to the fecond operation, I make
this fourth proportional a Parallel Sine of the greater fide of 70 gr.
© m.and take outthis Parallei Radius. For this meafured from go gr.
coward the Center, will be the verfed Sine of 21 gr.34 m.
~ Inthe like fore in the fame Triangle Z P S, having the fame Comple-
ments given, the Angle P ZS which is the Azimuth from the North
parcof the Meridian, willbe found 130 gr. 3 m. For here the Bafe
oppolitero the Angle required being 70 gr. and the difference of the
fides 38 gr. 30 m.and 40 gr. being 1 gr. 30m, the difference of their
verfed Sines will be che fame wich the diftance between the right Lines
of 20gr.and 88 gr. 30 m. This difterence I take and make ita Paral-
lel Sine of the lefler fide 38 gr. 30m. fo the Parallel Radius will be
the fourth proportional. Then coming to the fecond Operation, [.
make this fourth proportional a Parallel Sine of the greater fide 40 gr
andtakeout this Parallel Radius ; for this meafured from go *er. bes
_-yond the Center, in the Lines of Sines, ftretched forth at their ful!

length, willbe the yerfed Sine of 130 gr.3.m, °

2.U may find an Angle by knowing three fides, by that which I have
elfewhere demoni{traced upon Barth. Piti{cus,and that at one operation:

in chis manner.

Atthe Sine of the greater fide,
~ istothe Secant of che Complement of che other-fide: |
Sothe differerence of Sines of the Complemese of the Bafe,.
and the Ark compounded of the Ieffer fide withthe
Complement of che greater, .
co the verfed Sineof the Angle required..

So inthe fame Triangle Z PS, having the fame Complements given,,
the Angle at P, which fhewetlithe hour from: the Meridian, will be
found as before, 31 gr. 34 ™.

For the fides being 38 gr. 30 m, and 70 gr.O m, I take the Secane
of the Complement of 38 gr. 30 m. and make ita Parallel Sine of
7ogr. then keeping the Sector at this Angle, I confider chat the
Complement of 70g7. being 20 gr. added unto. 38 gr. 30 m. the com-
pounded fide (which isherethe Meridian Altitude ). will be 58 gr.
| j 39 7%.

cs

96 Of the Refolution of Spherical Triangles.

a

30 m. and thatthe Bafe being 40 gr. the difference of Sines of the
compounded fide, and the Complement of the Bafe will be (.as before)
the diftance between the Sines of Sogr.and $8 gr. 30-m.. Wherefore
I'take out this difference, and lay iton both ehe Lines of Sizes from
the Center: fo che Parallel taken inthe terms of this difference, and
meafured from 90 gr. towardsthe Center,doth give the verfed Sine of
Per, 34. : ,
4 This example of finding the hour of the day might otherwife have
been pro pofed in thefe terms, |

As the Sine of the Complement of the Declination,
isto che Secant of che Latitude : oe a at
So the differerence between the Sine of the Altitude propofed,
and the Sine of che Meridian Altitude, - $
to the verfed Sine of the hour from the Meridian : ;
Then the Latitude being 51 gr. 30 m. the Declination 20 gr. North-
be and the Altitude 50 gr. che work would be the fame as
before.

The other Angles PZ S, PS.Z, may be found in the fame fore : bur

°

having the fides and one Angle, it will be fooner done by that which

we fhewed beforein the 18 Prop.

20. Tofind a fide, by knowing the three eAngles,

If for the greater Angle, we take his Complement to 180 gr the An-
gles fhail be curned into Ades, and che fides into Angles, and the ope-
ration fhall be the fame, as in the former Prop. | |

As inthe Triangle ZPS, having the Angle ZPS 33
ZSP 30 gr. 28m andPZS130g¢r. 3m. Iwould take the greater
Angie of 13097. 3m. out of 180 gr. and there remains 49 gr.57 m.
Then as Ihad a Treongle of three known fides, one of 31 gr. 34 ms.
another of 30 gr. 20m, and athird of 497. 57 ™. I would feek the
Angle eppoliteco one ofthefe fides, by the Lait Prop. So the Angle

&'+ 34 ™.

_ which is thus found, would-bethe fide, which is here required,

21.70 find 2 fide, by having the other two fidesy and the Angle comprehended.

may be

This Propoficion being the converfe of the nineteenth,
. wroughe

two Rectangled Triangles, AC B, DCB. Then may we find AC the
m4! O

U bell
Seed

Of the Refolution of Spherical Triangles. . 97
wrought accordingly : but che beft way both for ic and thofe which
follow, isto refolve them into two Rectangles, by lerting downa Per-
pendioular, as was fhewed in che firft Prop. |

So inthe Triangle Z PS, kaving Z P the Complement of the Lati-
tude, and P Sche Complement of the Declination, with Z PS the An-
gle of the hour from the Meridian, we may find Z S the Complement
of the Alticude of the Sun. !

_ For having let down the Perpendicular ZR, by the firft Prep. we
have two Triangles, 2RP,Z RS, both reétangled acR. Then may
we find the fide PR, either by the fecond, or tenth, or eleventh Prop.
which taken ourof PS, leaveth che fide RS: withrchis RS andZR
we may find the Bafe Z S by che fourth Prop.

— Or having leedown the Perpendicular S M, we haveewO Redtangle
TrianglesS MZ,S MP. Then may we find MP, from which if we take
ZP, there remaineth MZ: but with MZ and SM, we may find the

- Bale ZS. ‘

22. To find a fide, by having the other two fides, and one of the Angles next
the inquired fide.

Soin the TriangleZ PS, having Z P, the Complement of the Liti-
tude,and DP S the Complement of the Declination, with PZS the Aa-

| gle of che Azimuth, we may find Z Sthe Complement of che Altitude
| of che Sun. 7

For having Z P, and the Angle at Z, we may to SZ ptoduced,
Jet downa Perpendicular P V. Then we have two Rectangle Tri.
angles PV Z, PVS, wherein if we find the fides VZ, VS, and

_ take che one out of rhe other, there will remain the fide required
SLs:

23. To find a fide, by having one fide, and the two Angles next the inqui-

red fide.

So inthe Triangle A B D, having AB the place of theSan, and

BA Dthe Angle of the greareft Declination, and AD B the Angle

of che Equator with the Horizon, we may find AD the Oblique

_ Afcention.

For having lec down BC the Perpendicular of Declination,we have

right

98 Of the Refolution of Spherical Triangles,
right Afcenfion, and D the afcenlior al difference; and comparing —
theone withthe other, thereremainehAD. | ~

24 To find a fide, by having two Angles, and the fide inclofed by |

them.

So inthe TriangleZ P S, having the Angles at Z and P, with the
fide intercepted Z P, we may find the fide PS, For having let down -
the Perpendicular PV, we have two Rectangles PV Z,P VS, Then
may we find the Angle V PZ,-eicher by che feventh, or fitreenth,
or lixreenth Prop. which added to ZPS, maketh che Angle VPS, —
_with ch’s-V PS, and P V,we may find the Bafe PS, accordingtothe
13 Prop. * :

25. Pofind an Angle by having the other two Angles and the fide inclofed

by thom. ° ; 7
| Sointhe Triangle ZP5, having the Angles at Zand P, with the
fide intercepted ZP, we may find the other Angle ZS P. For having —
lec down the Perpendicular ZR, we have two Re@angles ZRP, —
ZRS. Then may we find the Angle P ZR by the fixteenth Prop. and —
that compared with PZ S, leavechthe AngleR ZS: with thisR ZS; —
and Z R, we may find the Angierequired ZS R, according tothe fixth —
Propofition. q

26. To find an Angle,by having the othcr two Angles, and one of the fides 1
sext the inquired Angle. ‘ ’

So inthe Triangle A BD, having the Angles ac Aand D, with the —
fide AB, we may findthe Angle ABD. For having tec down the —
Perpendicular BC, we have two Rectangles, ACB, DCB. Then |
may we find the Angles ABC, DBC, andrake DB Coutof AB C39
tor fo there remaineth the Angle required A BD. i

27.7 0 fiad an Angle,by knowing two fides,and the Angle contained by them. © {

Soin the Triangle Z PS, having the fides Z P, PS, with the Angle
comprehended ZPS, we may find the Angle PZS, For having let
dewa the Perpendicular SM, we have two Re€tanglesSMZ, SMP,
| Then —

The Vfe of the Meridian Line. “99
Then may we findthe fide M P, and taking Z P out of MP, there re-
maineth M Z: withthis M Z and the Perpendicular M S, we may find
the Angle M ZS, by the fourteenth Prop. This Angle M ZS, taken our
of 180 gr. there remaineth P Z S. |

28. Tofind an Angle, by knowing the two fides nexe it and one of the other
Angles.

So in the Trisngle Z PS, having che fides ZP, and PS, withrhe
AngleP ZS, we may find the Angle ZPS; For having let downthe
Perpendicular P V, we have two Reétangles P V Z,P VS. Then may
we find she Angles V PZ, V PS: and taking VPZ out of VPS,
there remainech Z P S, which was required.

Thefe 28 Cafes are all chat can fall our in any Spherical Triangle :
if any do. not prefenctly underftand them, lec chem once more read
over the ufe of che Globes, and they fhall foon become eaiie unto
chem. me i

CHAP. VI.
of the Ufe of the Meridian Line in Navigation.

y= Hie Meridian Line is here fer on che fide of the Seétor ftrecched
er at full Jengeh, on the fame Plane with the Line of Lines and
Solids, and is divided unequally toward 87 gr. (whereof 70 gr. are
about one half ) infuch fort as the Meridianin the Chart of AMereators
— Projeétion. The Ufeof it may be, |
"1. Te divide a Sea-chart according to Mercators ProjeEiton,
If a degree of the Equator on the Sea-chart, be equal co the hun-
dred part of the Line of Lines in the Sector, the degrees of the Meri-
dian upon the Seétor, fhall give the like degrees upon the Sea-chart:
if otherwife they be unequal, then may the Meridians of the Sea-
chare be divided in fuch forcas the Line of Meridiansis divided on the
Seéto® by that which we fhewed before inthe 8 Prop. of the Line of
Lines, é

~ Burto avoiderror, I have here fet downa Table, whereby the Me-
ridian Line may be divided out of the degrees of the Equator fu ppoting

O 2 eic

too —s«THE Uf of the. Meridian Lint.
each depree inthe Equator , to be {ubdivided into a thoufand parts.
By which Table, and the ufual Table of Sines, Tangents, and Secanrs,
_ the Proportions following may be alfo refolved Arithmetically, For
the manner of divifion, let the Equator be drawn, and divided, and

croffed with Parallel Meridians, as in the common Sea-chart: then —

look into the Table, and lee che diftance between the Equator and
40 gr, inthe Meridian, from che Equator, be equal to 43 gr. II parts

of che Equator, as in the Table: let 50 gr. inthe Meridian trom —

the Equator, be equal to §7 gr. 909 parts of the Equator, and fo in
the reit. wT,

The making of th's Table is, by addition of Secants. For the Para!-

Jels of “Latitudes being Jefs than the Equator or Meridian in fuch pro-—

portion as the Radius isto the Secant of the Paralle). For example,
the Parallel of 60 degrees of Latitude is lefs than che Equator (and
confequently, each degree of this Parallel of 60 degrees lefs thana
degree of the Equator, or Meridian) in fuch proportion as 100000 the
Radius, hath unto 200000 the Secant of 6o degrees. |

»
~

A Table for the Divifion of the Meridian Line. 101 |

Se

Ms: Gr. Par. [At Gr. Par.

O

oHatey Gey

| ree,

1900
I.000| 4
Pitels| Pe

3.001
3.101
3.201
3-302

eee fe

f

M|Gr. Par.|M|
sO.0 FI
6,111
6.212
6.33 172

“6

——

Ed

eee

Gr. Par.| A| Gr. Par,

‘ 9.037|12 22.058
ee 12.19¢
9.239), | 12.293
9:341| loo a
9.442 12 407!
PDAS 51) 2.00
9-045 12,702
9.740 12.805] >
9 848 12.907
9.949 Hz.016

‘10.051 I2| 13.112
TO.1j§2 13.215!
10.25 4 137,318
10.35 § L342 1
10.457 13.523
10.550] | 12,626]

LO.06i | = 13.729| 4

10.702 13.332
TO.S64) | 13.935
10.966 14.03 8

| 11.068/14} 14.141}

11.170 14.244

T1272 14.347]
11.374 14.450

11.476 er
11.578 14.656
EL OBOE 14.700
11.782 14.563
11.884 14,967 <
11,986 15.070
12.088

I§! 15.174

|ro2z ATable for the Divifion
M Gr. Par.|MM\ Gr. Par. M)\ Gr. Par M| Gr. Gr. Par,

~

M Gr. Par,

15|_15.174|18] 18.303 21| 21 486 24 “24.7 24: 27| 28.058
15.277 18.408]. | 21.503 p24 Bay 2¢.171
15.281 18 513 21.70 24.953 28.28 3
15,485 18.619 21.808 25.06} 22.396
15.588} | 18.724 21.915 25.173 28.508

I 5.692 18.830 21.023 25 282 32634
15.7961 118 935 226130 23.392 22.7 34

1§ 900 19.041 22.236 25.502 28.347
16.004 19.146 22.345 23.613 28.959
16.107 19,251 22045 3 25.723 Pl esiiy we:

16| 16-211} 19] 19. 356122] 22.561)251 25-833128] 20.186
16.316 19.403 22.669) | 25.943 29.299
16.420 19.509 22.777 20.054 29.313
16.524 19.675 22.885 26.164 29.526
16,628 19.7811 | 22.9931 | 26.275! | 29.646

_! 16.732 19.887] | 23.101| | 26.386 29.75 3
16.836] | 19.693 23.210) | 26.497) | 29.867
16.941 20.100 24.318 26.608 29.981
17.045 20.206 23-427 20.719 30.095
17.150 20.312 232935 26.830 30.209

17| 17.255 |20] 20.419]22| 23.643/26) 26.941]29] 30.324
17.359 20.525 23752 27.052] | 30.438
17.464 20,6 32 23.8611 | 27.164 30.553
17.568 20 738 23.970] | 27.275 30.667
17.673] | 20.845] | 24.070] | 27.387 30.782
17.778 20.95 2) _ 24.188 _| 27.499 30.897

~~ | 17.8831 | 21.059 24.297 27.610! | 31.012
[7.988 21,155 24.406 270722 31.127
18.093) | 21.292] | 24.515] | 27.8341 | 31.242
18.198 | 21.379} | 24.624) | 27.946! | 31.357

13} 18.304|21! 21.486!24 24734127! 28, os8i3ol 37, 473

:
4
j
Q
|
|
4
a
i
;
f

— . »
SS nS os

i

34.873

38.509

42.287

42! 46.362145! 50.49

| 3.4.992136! 38,63 3139! 42.415 : 499)

of the Meridian Line. 103
| ag| Gr. Par | AA| Gr. Par.| A4| Gr. Par [AA] Gr. Par.|AL| Gr. Par.
316473132] 34-992]36] 38 637] c} 42-4! sak 46 362
31.885] 1 35-r11} | 38.7571) 42.544] | 40.496
21.704 3§.231T | 38 880 42.073 46.631
31.820 35.350] + 39004[' | 42,802 46756
31.936 35.470 39 [29 42.931 4.792;
32,052] | 35-590 ah : a bis
37.1681 | 357101 1 39-377 43 191 Lh i
2.284 35.830 | 43+320 47 +309
32.409 35-9590 3.926027 43-452 47 044.0
32.516 36.071 39. Bal) 43.581 47-581
32.633 34 36.191137] 39.877 ie 43.711123| 47.718
32.750 36.312 40.002 439842) | 47.855
32.867 36.433 40.128 43.973 47.992
eck 30.5 $4 40.258 44.194 48,129
SGOT LLL 30.675 40.379 44.235 48.267
ee st 36.796] | 40505 44.366 48.404
ra 36.917 40 631] | 44-498 ; 48,54.2]
33.453)- 1 37.0390 40.757 44 030 48.681
33-571] | 37-161 40 854 44.762 48.819
33.688 37.282 41.014 | 44 894 48.958
33.806}25| 37.405 38) _41.127/41} 45-026l441 40.070]
33.924 37-527 41.264 45.459 49.236}.
34-042 37.643 41.322 45292 ra
34.161 37:771 41.519 45-425 49.5158
34.279} | 37-8904) | 41.646) | 45-558] | 40.655
34-397) __| 38.017] | 43-774) | 45-691] | 49.705
34.5161 |» 38.140 45.8251 | 40.035

50.358}

fa

} A Table for the Divifion -

M| Gr. Par.|M| Gr. Per.| A4| Gr. Par. M Gr. Par.| A4| Gr. Par.
45) 5429} 48 ¥4-800]57] 59 481 54| C4eg12 57. 69.711
} 50.041] | §5-O10}~ | $9.040/"" | 64,582] | 69.805
| 50.7831 155-260] 4 59.800 64.7 53 701080
| $0925] | $5.31 59 960 64.924 70 265
51.068 55-460} | 60.120 65.096 70.449
SEALS OL. 60.280 _ | 65.268 70.035
51.353] 1 35-7621 I *60.g4il 165 gaol” | oo gay
§ 1-496 55-913 60 601] 65.613 71.008
51.039 $0,065) . 60.763 65.786 71.19 |
51.785 $0.21) 60.92§ 65.960 “tae,
46|_51.977/49! 503091521 O1.088l551 66,1341 68) 71.592
§ 2207 i 56.522, arora 663308 ee 71.701
way i 56.675 61.413 66.483 71.950
52.360 50.328] | 61.577 60.659 3 Gad
5 2.505 56.981 61,741 66,835 ae
| 52-950] 37-135) ) 61-904 87.011 72922
52.79 § 57.289 62 669 OZ ORS ons ie
§ 2.941 57-444 62.234 ) 67.365 i

| 53.087 37-598 62.399 67.543 a
§ 3.233 57-704 62.564 67.721 vt ae
7|_52-350/59} §7,.909/53| 62.730 56] 97-900 5 fee
Gila? 225° =| £27190] 201591 73 486
53. 26 BROOS) 62.897 68,079\5 fay eee
| $3673 58.221) 63.063 68.258 hed
1 | 53.8211 -1§8.377] | 63.2311 | 68.438) | 73°97
53.968 38.534 63.398 68 618 sei i
| 54.116] 58.691 __| 63-566 68.799 ora
5 4.204 58 348) | 93.734) — 68.981 us 4 +
54-413 §9.000 03.903 69.163 a
54-$02) |} 59.164 04072 69.345 gs Me
34.711 590322 04.042 69.528 | pita 2
148 §4-800151) $9.481'54 64 412'57 69.712!60 eee

75-456

M

{60

105
Gr. Par.|A4| Gr. Par.|A4 Gr. Par.| | Gr. Par, M)\Gr. Par.
75-451103| 81.749/66| 88.725 |05| 96.575 72|1053579
75.656| | 81.970 88.971) | 96.854) [105.904
75.85 3 82,191 89.219 97-1351 106.230
76.057| | 82.413| | 89.467] | 97-418) |106.558)
76.261} | 82.635) | 89.716| | 97.701] {106,888}
-76.464| | 82.860] | 89.967 97:986| _|107.220

~ | 76.6671 | 83.0841 1 90.218 98.272! 1107.5 43
76.871 83.313] |-90.470 98.560} 107,888
77.076 83.536 90.723 98.849} {108.226
77-281} | 83.7631 | 90.978] | 99.139] {108.6651
77-487164|_83.990167) 91-232\90| 99-4 31172/108,906
77.694| | 84.219 91.489| | 99.7241 |109.249
77-901, | 84.445} | 91.746) [100.018] 109.504
78.109 84.678] « | 92.005}. [100.314 109.941
78.367 84.909} 92.264] {100.612! |r10.290
78.526 85.141 | 92-525 100.910} |110,647
78.736) | 85.374 92.787} |101.211| |110,.904.
78.947 85.607 93.050] 101.513] farr.349
79158 85.842 93-314. 101.816 111.707
79.370 86.077 93-579| |1O2.121 [112.066
79.583165| 89-313|68) 93.846|71)102.427| 541112. 428
79.796| | 86.550 94.213] |102.735 | 142.702
8o.o1ol | 86.788 94.382! .1103.044 biseerea
80.225| | 87.027 94.652] [103.356] [113.526
80.441 87.267 94.923} |103.668| |113.897
80.657 87.508 _.| 95-195] |103.983! |114.270
80.8741 | 87.749! | 95.468 104.209\— 114.645
81.091 87.9921 | 95.743 104.616] [115.023

81.310 88.235 96.019] {104.936} 115.403
81.529 88.480 96.296} 105.257] 115.786

of the Meridian Line.

631 81.749166! 88.72 5!169! 96.575172

10§.§79'75'T16,171
P

ee ee

AM Gr. Par MM Gr.Pare MéM Gr.Par. M| Gr. Qr. Rar, M| Gr. Par.
Lye lri6.t74 78 129.075 |87| 145-650 84 1086947 87 208 705
Ue 116.559] 1129.5 58] |14.6,292 169.912 ao

i 06 A Table for the: Dizsfir iomof the Meridian Lines |

116.9491 [130.065 146 942] |170.893 212.668

[117.342] [130.536] 1147600] l171,891| [ar4.745

117.737 ieee 148.265 172.907 216.909
F bea 131.930 148. 937) |173.941 219.158
H

118.7301 [132.034] |14o. 149.618] 1174994 221.498]
118.939 132.542 1-§0.307 176.067] |223.938
119.345 13 3.055 151,003 177.100] |226.486!
f19.755| °|233-572 151.709 178627 5 229.153
76|120.166|79|134-094)82 152 4231851179.411188123 1.950]

“[aates Bi 134.020 (53.147 180.569|— 234.891
4 |i2zzoco} [135-151] 11536878) |181752) [237.9914
| 121.420| |135.687} |i154,6206| [182.960], 241.268] —

(27.843 136.2281 [195.3721 .1184.1941)
1123,270] |£36.775] |156.132] 11954454
1223700). |£3'723261]. [156.903 |/- 1 96.743|T 1525
P23 153 3 137.883], |157:685| |188:062/|
|r2g.g70l 'r3B.4q5!! l1y8.478! M89.4r11

1¥24.009 T3012 159.281 ligo, 793

77|124.4§2|80| 139.585 |83|160.096|8 6/192 210 89 2714705}
Gon 140.164) }160.922]~ 193,661] |277 76,
(25.340! 1140:748! 1 161.761

195, 191
162.612 196.680
163.474] |198.25§1 301,058}
164.3252] |199.867 311.5621)
1O§ 243, (201.529) 1324 455}
166.146 203.240 341.166
167:065 205.00 5 365.039
128,596] |145.014 [167 999} |206.825] “|408.0r1

[78 £29.07 5181} 145 650!84'168.9471871208.705 90! Infinite: |-

284,517]

|r25.501 141.339) 292.191

‘
| 1126.258) |141.936

1126718) 142.538

er Pak PRAT ST
Rijs WRF ARSED N= *
DOr

OHO 4

WwW W

w

wa WW

143.1471 |
127.649 £4.3.763{
r28.121| -|144.385

107

108 The Vfe of the Meridian Line.

If i¢be a particular Charr, I would firjt draw che Line A B fer-
ving for the firit Meridian, and crofs ic wich cwo Perpendiculars BC
and A D, the one at the upper end, the other at che lower end of che
Coart, whica may ferve for the extreme Paralleis of Latitude chat you
arero mike ule of. i

Taen conlidering at what Laritude che Chart is ro begin and end,
andthat chis Chare intended for che Latitude ot chefe parts, is to be-
gin at5ogr. and fo end at §5 gr. I look intothe Table, and find chae
50 gr. of Latitude muft be drawn at §7 gr. 909 parts ; and 55 gr.of La
citude at 66 gr. 134 parts from the Equator; and that the Meridian
diftance between rhe Parallelof 50 gr. and 55 re of Latitude matt be
equalto 8gr. 225 parts of the Equator. Whereupon I take che Line
A Bout of the Meridian Line, and diminifh it in fuch proportioa as
8.225 hath unto 1000 per 3 Prop. Line and with that extent of the
Compafies, I divide the ewo extreme Parallels of Latitude into equal
degrees, and through each degree draw Meridian Lines parallel to che
firft Meridian, noting them with 1, 2, 3, 4, cc, and then, I fubdivide
eicher one or all of thofe degrees into ten parts, and ( if I may ) each:

renth pare into ten partsmore, bute howfoever, I fuppofe each depree = |

to be fubdivided into 1000 parts. |
The Meridians being drawn, I come to the Parallels of Latitde,
beginning at 50 gr. - | | |
And finding inthe Table, that che diftance beeween the Equator and
for. in the Meridian fhould be equal to 57 gr. 909 parts in the
Equator and his Parallels, I may fuppofe the lowett Parallel to be
57 gr. trom the Equator; fo theidiftance between this lowe(t Parallel
and che Parallel of 50 gr. will be qnly 9e9 parts, Wherefore I take
thefe 909 odd parts, ouc of che degrees that I divided before, and
prick them down into the ewo utrermoit Meridians from the loweft
Parallel upwards, and there draw the Parallel of 50 gr. of Lati-
tude. e |
_ Inlikemanner, becaufe I find by the Table char the diftance be-
tween the Equator and 51 gr. in the Meridian is 59 gr. 881 partsof
che Equator,I abate the former §7 gr. and thereremain2¢r. 48t parts
ior the diftance between the loweft Parallel, and this Parallel of 51:
_ wherefore] takethefe 2 degrees 481 parts oucof theLinebefore diz
vided, and prick them down in the two uttermoft Meridians (as be-

fore ) from the loweft Parallel upward, and there draw the Parallel of :
gi deprees of Latitude.)

If

2,

“rT0 The Ufe of the Meridian Lint. :

If. any defire to have hisChare agree with his Sector, hemay make
each degree of Longitude equal to the tenth pare of the Line of
Lines, and divide the Meridian of his Chare out of the Seétor: fo
fhall ‘each degree ‘of the Charc be ten times as large as the like
depree on the Seétor, and the work be eafie from the one tothe
other, , :

Or he may divide the Meridian of his Chart by the fide of a Pro-
tractor, ‘fuch as is commonly ufed by Surveyors of Land, and is here
reprefented by AC DE, wherein the outward part of the Semicircle
A B Cis divided-equally into-180 gr. The inward part-equally into
16 Rumbs, and each Rumb fubdivided into 4.

4.

7 ;
a :
FOUN eT

ft al CT Toe Ly D

anadGuaREApotpyDy
Po TOSSEVUARBABSTAUSPESETIDEGES BOE: PPT Petit tty s

PRDUADOUBUDNCRIEATINGS

The Ofe of the Meridian- line, 111:
The Lines C D; DE, EA, divided equally according to the Line
of Linesupon the Seézor, or the Parallelsupon theCharr. Onely the ~
Diameter A.C would! be divided. unequally, by letting down occule
perpendicular Lines upon it from. each Degree in the Semicircle ,
which being done, the intermediate parc between, the- Rumbs ao
the Diameter may be all cur forth: And the: back fide of the-long
Square may be filled. with 6 Lines of Chords, or Scales of feveral -
parts in the Inch. ; ri
So may the Meridian be divided by the parts. of the Side E D,
the Angles of cacli Rumb may readily be pricked down: by the De-
grees in the Semicircle, and the: Line of Chords;and the other Scales
may ferve to do the: like with more variety.

ad

2, To find how many Leagues anfwer toone Degree of Longitude

' in every feveral Latitude.
In failing by.the Compafs, the Courfe holds fometime upon’ a Great
Circle, fometime upon a. Parallel co the Equator; but moft commonly
upon crooked Lines, winding towards one of the Poles, which Lines are
well known by. the Name of Rumbs. |
If the Courfehold upona Great Circle; it is either North or South,
under fome Meridian, or Eaft or Weft under, the Equator, And in
thefe Cafes, every. Degree requires an allowance of twenty Leagues ;. eve-
-__ ry twenty. Leagues will make a. Degree difference ia the failing: fo chat
here needs no further Precept than the Rule of Proportion in the Chapier

of Lines. .
But if the Courfe hold Eaft or Weft, or any. of the Parallels co the

Equator,
As the Radium, |
is to. twenty Leagues, the Mdeafure of one Degree. at the Equa-

tor :
So the Sine of the Complement of the Latitude, |
to the <Meafure of Leagues anfwering to one Degree in that
Latitude,

Wheres

The Vfe of the Meridian-line. a
Wherefore I take 20 Leagues out of the Line of
Lines, and make it.a parallel Radius, by fittingic over’ —
in the Sines of go and.go: fo his parallel Sine caken
out of the Complement of the Latitude, and meafured
in the Line of Lines, fhatl fhew the number of Leagues
required. : ‘

Thus in the Laticude of 18 gr. 12m. we fhall find
19 Leagues ah{wering to one Degree of Longitude, and

.18 Leagues in the Latitude of 25 gr.15 m. as in this
Table. -

This may be done more readily without opening the
Sector, by doubling the Sine of che Complement of the
Latitude, as may appear in the fame Example,

Ic may alfo be done by the Line of Meridians, eicher
upon the Sector, or upon the Charc: For if we open'a
pair of Compafles to the quenticy of one Degree of Lon-
gicude in the Equator, or one of his Parallels, and

*meafure ic in the Meridian-line, fecting one Foor as
much above the Latitude given, as the other falleth bee
neath it, fochat the Lacitude may be in the middle be-
tween the Feet of the Compaffes, che number of Leagues
intercepted thall be that which was required. |

But if the Courfe hold upon any of the Rambs, be-

tween a Parallel of the Equator and the Meridian, we are to confider

(befides the Equator of the World to which wetend, which muft be al-

ways known), |
1. The difference of Longitude, at leaft in general.
@. The difference of Latirude, and that in particular.
3. The Rumb whereon the Courfe holds. |
4. The diftance upon the Rawmb, which isthe diftance which weare

here co confider,and is always fomewhat greater than the like diftance up-

o greater Circle.And for thefe,firft,I fhew in general this third Propo-
1010Ne .

3.7¢

3. To find how many Leagues do anfwer to one
Degree of Latitude inevery feveral Rumb. ~

The Seamans Compafs is commonly divided
into 32 Points; the half, into 16; che quarrer,
into 8; which have their names of N NOE,
NWN E, es. according to thofe parts of the
World to which they point. Anfwerable to
thefe Points, are the Rumbs upon their Chare ;
each quarter divided into 8, each Rumb 11 gr.
85 m. diftant one from the othei: The firft Rumd
being that which is 11.gr. 15 a, diftance from the
Meridian’; the fecond, 22 gr. 30 m. the third,
33.gr.45 m, and fo the reft. And (if they have
need of {maller paris) they {ubdivide each Rumb
into quarters, allowing 2 gr. 48 m, tothe firft

guarter, 5 gr. 37 co the half Ramb, ec. as in |

the Table following.
As the Sine of the Complement of the Rumb
from the Meridian,
. isto 20 Leagues, the Meafure of ome Degree
of the Meridian: |
So is the Radius,
to the Leagues an[wering to one Degree xpon
the Rumb.
Asifin failing WE 5 N,from 50 gre of North
Latitude, it were required how many Leagues
the Ship fhould run before it could come to
51 gr. of Latitude, becaufe this is the third
Rumb, and the Inclination thereof 33 gr. 45 m.
T would take 20 Leagues, ec. rae ipe
Wherefore I'take 20 Leagues out of the Line
of Lines, and make ic a parallel Sine of 56 gr.
15 m. the Complement of the Rumb from the

_ Meridian; fo his parallel Radius taken and mea-
fared in the Line of Lines, -thall fhew me 24 for

the number of Leagues required.

Kes

og | Luclinat.| Number
of
ide Salis

Lgs.Pa’,.

S| to the
Merid.
Gr. M

It 151 20 39
I4 4.
p {tO §2
| E 411.
2422, 1301237 165
| [25 To
be [23 07
| [30 §6
313345
36 34
39. 22
4% 1
4:45 0
37 49
50 37
| |53 26
ny 56 If
152 4] 38 go
| JOl $2] 42 43
| [04 41) 46 78
| 6,97 30). 52 26
| |79 I9| 59 37
73 .71 68 90
Pr aleZ). a 8G 82 31
17178 4§|102 52
Bi 34/136 30
84, 22,205 24
|. (87 111407 60
| 8i90 ~— of [nfinita.
And

14

~ And thus in the firft Rumb from the Meridian we thall find 20

The Uje of the Meridian-line,

Leagues 39 parts anfwering to one Degree of Latitude, and 21 Leagues
65 parts inthe fecond Rumb, &e. asin this Table, where we fubdivide
each League inroa hundred parts, and fhew befides what Inclination the
- Ramo hath co che Meridian.

nA R2SARAsRERRAA A Ac ReeaAsetAAe AeA SnaemeLehase

ee oO)

te] ~ >

70|__' 80} (9°)

60

This may be done more readily wichout opening the

Seftor, by dou ling the Secant of the Rumb, as may ap-
pear in the fame Example.
_ Te may alfobe done upon the Chart, if firft we draw the
Rumb ; then we take the diftance upon the Rumb between
cwo Parallels, and meafure it in the Meridian-line, as
far above the greater Latitude as beneath the Ieffer. For che
number of Leagues intercepted fhallbe thac which was re-
quired. |

For Example: In the fecond: Chart, pag. 113+ I firft
draw the 8 Rum)s, from the InterfeGtion ot the Meridian
with the Parallel of 50 gr. of Lacicude, either by thac
which I have fhewed before in the general Ufe of Sines,
Chap. 11. Prop. 10. or by help of the Procraction laft men-
tioned; For, laying the Center of the Protractor to the
Point of Interfection ( which is co be the Cencer of the

Rumbs) and tarning the Diameter of the Procra&or until

it be parallel co the Meridians of the Chare (which is chen
done, when the Meridians and Parallelsin the-‘Chare fall

under like divifions in the Protra&tor) I may make one

prick at 11 gr. 15 7. another at 22 gr. 30.5. im the out-
ward part of che Semicircle;: and fo the reft.

Or, having neither SeGtor nor ProtraGtor, I would have
a Line of Chords.fet on che fide of the Ruler which I am
toufe, from which I may take 60 gr. and with that extent
{etting one: Foot. of the Compaffes in. the former Point of
Interfection, draw an ‘occult Ark of aCircle, and therein
prick down the: former Arks from the Meridian , as in
Chap. tt. Prop. to. So thefe Arks being pricked down by

either of thefe: ways, che: Right Lines drawn through the

Center and:tho fe pricks, (hall be the Rumbs required.
- The Rumbs being drawn, I take the diftance beeween

the Parallels of 50 and §t gr. upon:A.Cy the third Rumb ;

and

on ae

>

The Ufe of the Meridian-lines bee
and meafuring it in che Meridian-line, I find the Compaffes to reach
from above 75 of a Degree below the Parallel of 50, but 25 above the
dn of $1 gr. intercepting I gre z= or 24 Leagues, fuch as 20 make
a degree. 7

Kenita I take the diftance upon the fame Rumb between rhe Pa-
rallel of 54 and 55 gr. which I find to be fomewhat longer than the
former diftance berween the Parallels of 50 and 51 5 but meafuring
it in the Meridian Line, according to the Latitude of the Parallel,
I find but 1 gr. 7, (or 24 Leagues) as before, for the number of
Leagues anfwering to 1 Degree of Latitude upon this third Rumb,

And by the fame reafon, I may find the number of Leagues an-_
fwering to a Degree of Latitude upon the reft of the Rumbs aerce-
able to the Table,

This confidered in general, I fhew more particularly in twelve
Propofitions following, how of thefe four any two being given, the
other two may be found, both by Mercator’s Chare, and by this
Sector.

1. By one Latitude, Ramb, and Diffance, to find the
Difference of Latitudes.

As the Radius,
to the Sine of the Complement of the Ramb from the cHe-
ridian :
So the Diftance spon the Raesh,
to whe difference of Latitudes.

Let the Place given be A, in the Latitude of 50 gr. C in a great-
er Latirude, but unknown, the diftance ye the Rumb being 6 gr.
berween them, and the Rumb the third from the Meridian.

Firft, I cake 6 gr. from the diftance upon the Rumb, out of the
Line of Lines, and make ic a Parallel Radius, by pucting ic over in
the Sines of go and go: Then keeping the SeGtor ac this Angle, I
take out the Parallel Sine of §6 gr. 15 ms. which isthe Sine of the
Complement of the third Rumb trom the Meridian, and meafuring
it in the Line of Lines, I find it to be 5 gr. and fuch is the diffe-
rence of Latitude required. ‘3 : dads

Q 2 Or,

4

116 The Ufe of the Aleridian-line,
Or, I may cake out the Sine of 56 gr. 15 m. for the Comple-
ment of the third Rumb from che Meridian, and make it a paral-
Iel Radius; then keeping the Sector at this Angle, I cake 6 gr. for
the diftance, either out of the Line of Lines, or any other Scale of
Equal parts, or elfe out of the Meridian Line, and lay ic on both
fides of the Seftor from the Center, either on the Line of Lines or
Sines: fo the Parallel taken from the Terms of this diftance, and
meafured in the fame Scale wherein che diftance was meafured, fhall —
fhew the difference of Latitude co be 5 gr. as before. q
Bat in fhorter diftances, fuch as fall within the compafs of a days
failing, this Work will hold much better; as may appear by com-
paring the Work wich che Table following, where the Numbers in
_ the front do fignifie the Leagues; thofe in the fide, the Rumb; and
the reft in the middle, the difference of Latitude. ,
In the Chart let a Meridian AB be drawn through A, and in
_ A with AB make an Angle of the Rumb BAC: Then open the
Cornpafles, according to the Latirude of the Places, to EF the quan-
tity of 6 gr. in the Meridian, transferring them into che Rumb from
A to. C, and through C draw the Parallel BC, crofling the Meri-
nian AB in Bs fo the Degrees in the Meridian from A to B hall
-thew the difference of Latitude to be 5 gr.

a. By the Ruwb and both Latitudes to find the Diffance
xpos the Rumb.

“As the Sine of the Complement of the Ruwsb from the Meridian,
ts to the Radius: , |

| «So is the Difference of Latitudes,
to the Diftance upon the Rumb.

As if the Places given were A in the Laticude of 50 gr. Cin
aif Laticude of §5 gr. and the, Rumb the third from the Meri- |
dian, | id . a
Here I may take 5 gr. for the difference of Laticude out of the
Line of Lines, and put it over in the Sine of 56 gre 5 m, for the

; Com-

a epee

s ‘rz

| 100 | 80 | 60 | 40 20 ro bro pra) 6b ast |

The V fe of the Meridian-lines
AT able of Leagues, Rumbs, and Difference of Latitude.

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The Ufe of the Meridian Lines. Tg.
Complement of the third Rumb from the Meridian. . Then keeping
the Sector at this Angle, I cake out the Parallel Radius, and meafu-
ring isin the Line of Lines I find it to be 6 gv,and fuch is che diftance :
nponthe Ramb, which was required.

Or I may take the Lateral Radius, and make ic a Parallel Sine of
§6.gr. 15 m. the Complement of che Rumb from the Meridian: then
keeping the Se&or at this Angle, Icake 5 gr. for the difference of La-
ticude, either out of the Line of Lines, or out-of fome orher Scale
of equal parts, and lay icon both fides of the Seétor from che Center,
either onthe Lineof Lines or of Sines: {0 the Parallel taken from:
thecerms of his difference, and meafured inche fame Scale with the
the difference,thall thew the diftance upon the Rumb to be 6 gr.or 12.2
Leagues. ahs

Or keeping the Seétor at this Angle, I may take the difference be-
tween gO gr. ands 5 gr. out of the Meridian Line, and meafuring ic
in the Equator, I fhail find irto be equal to 8 gr. 229, of che Equa-
tor. Wherefore I cake the Parallel between $22 and 822 out of the
Line of Lines, and meafuring it in the Line of Lines, 1 fhalkfind icto-
be 989; which fhews that according to this-proje€tion, the diftance ~
uponthisthird Rumb, anfwerable tothe former difference of Lati-
tudes, will be equal to 9 gr. 89p. of the Equator. . ;

Or the Seétor remaining at chis Angle, I may take the diference.
between 50 gr.and 55 gr. outof the Meridian Line, and fay it from.
che Center on both fides of the Seftor, either on the Line of Lines or
of Sines: forhe Parallel caken fromthe terms.of this difference, thal
beche very Line of diftancerequired, che fame with A C or EF upon.
the Chare; which may ferve tor the better pricking down of the di-
ftanceuponthe Rumb, without taking it forth of che Meridian Line,as.
inthe former Propoftion.

Or if che Rumb fall nearer to the Equator, thatthe lateral Radius.
cannot be ficted over init, this Propofitionmay be wrought by Parallel
entrance, |
_ For,if Bfrfttake out the Sine of 56 gr.15 mand make it a parallel
Radius, by fittingisover inthe Sines of 90 and99, or -in the ends of ;
the Line of Lines, and thentake 5 gr. for che difference of Latitudes ;
our of the Line of Lines, and carry ic parallel to ehe former, I hall find
ieto.crofs both Lines of Linesin the Points of 6: .andfo it gives che.
fame diftance asbefore.... . |

Or if. che diftance:be-fmall, it:may be found by the former oe
Or:

2
Pia

_ For the Rumb being found in the fide of the Table, and the difference

128 The Use of the Meridian Lines

of Latitude inthe fame Lines che top of the Columnwherein the dif-
ference of Latitude was found, hall give the number of Leagues in the

diftance required.

- Or we may find this diftance in the Table of Rumbsin the fifth Pro-
pofition following. For according to the example,look into the Table of

' che third Rumb for 5 gr. of Latitude, and there we fhall find 6 gr.10

parts under the title of diftance.

So if the difference of Latitude upon the fame Rumb were 50 gr.
the diftance would be 60 gr. 13 parts. If the difference of Latitude

«pon the fame Rumb were only 4 of a degree, che diftance would be |

only 60 parts, fuch as 100 do makea degree.

: Inche Chart leta Meridian A B be drawn through A, ard Parallels
of Lacitudethrough A and Cs; and then in A, with A B, make an
Angle of the Rumb BAC: fo che diftance take from A to €, and
meafured in the Meridian Line, according to the Latitude of the pla-
ces, fhall be found to be 6 gr. or 120 Leagues. And fuch is the diftance

_ required.

3. By the diftance and both Latitudes, to find the Rumb.

As the diftance uponthe Rumb,
to the difference of Latitudes :
So is the Radius,
to the Sine of che Complement of the Rumb from the Meridian.

As ifthe places given were A, in the Latitude of 50 gr. Cin the
Latitude of 55 gr. the diftance between them being 6 gr. upon the
Rumb. Firft I take 6 gr. for the diftance uponthe Rumb, and lay icon

both fides of the Sector from the Center ; then out of the fame Scale —

take 5 gr. forthe difference of Latitude, and to it open the Seédor
imthe terms of the former diftance: fo the parallel Radius taken and

meafured in che Sines; doth give 56 gy.15:m. the ‘Complemene where-
of 33 gr.45 m. is the Angle of the Rumbsinclination cothe Meridian,
which was required... | saith l ad:

In the Chart let a Meridian A B be drawn shrough A, and Paral- 1

lelsof Latitude, both chrough A and C:; ‘then: oped the Compatve<
according tothe Latitude of the places to E ip 9 of pipe

_ the Meridian, and fetting one foot in A,’ turn che other till i¢crofe the

Parallel

Be, de

eet ; : 3 ( ; fen. ‘ . 5 4 “

_ ‘y ‘) + 4

| The: Ufe of the Meridian Lines 121
Parallel B Cin C, and draw the right Line A C: fothe Angle BAC
fhall fhew the inclination of the Rumb to the Meridian to be 33 gr.
45 mas before. : Re ELES |
Thefe three laft Propofirions depend one on the other,-and may be
wrought as truly by the Common Sza-Chart as by this of Afercators
ProjeGtion : and therefore in working them by the Seéfor, the diftance
_and the difference of Latitudes may-as well or better be taken out of
the Line of Lines (which here reprefenteth the Equator) or any
other Line of equal parts, as out of che inlarged degrees in the Me-
ridian Line. But inthe Propofitions following,che difference of Lon-
situde muft be taken out of che Equator; the difference of Lati-
tudes and diftance upon che Rum) muft alwaies be taken one of the

Meridian Line: which I therefore call the proper difference,and pro-

per diftance. |
4. By the Longitude and Latitude of two places to find the Ramb.

As if the places given were A, in the Latitude of so gr. Cinthe
Latitude of s5 gr. and the difference of Longitude between them
were 5 gr. 30 ™, ?

In the Chart fet Meridians and Parallels be drawn through A and
C, and a ftraighe Line for the Rumb from A to C3 then by that we
fhewed Cap. 2, Propofition 9. inquire the quantity of the Angle
BAC, and-it fhali be found to be 33 gr. 45 m. which is the third
Rumb fromthe Meridian. Wherefore the proportion holds for the
Sector, |

As A Bthe proper difference of Latieude :
isto BC the difference of Longitude:
So is A B Radius,
to B.C, the Tangent of the Rumb from the Meridian. -

According to this I take the proper difference of Latitude from
50 gr.to §5 gr. outof the Line of Meridians, and lay it on both fides
of the Seéter trom the Center ; then I take the difference of Longi-

tude ser. 4 oucof the Line of Lines, andto it openthe Sector inthe ©

terms of the former difference of Latitudes, fo the Parallel Ra-

dius taken from between 90 and 90: and meafured in the greater —
R

Tangent

eo os ~ Be eee = > rs _

eae

waz” The Uf of the Meridian Linen,
a Tangent on the fide of the Seétor, doth give 33.¢r. 45. #. forthe Rumb.
fe. 4S required.” ‘ Te hig dee i

Mi Butif the Rumb fallnearer to the Equator 5.

uy)

As A Dithe difference of Longitndes,.
is to D.C che proper difference of Latitudes :.

ba So A Dehe Radius, "i
iy co D-Cthe Tangent of the Rumb fromthe Equator.
: Be According-to this Itake the former difference of Latitudes from

Fa.gr.to 55 gr. out of the Line of Meridians, and to it open che
Seétor. inthe terms of the difference of Longitude reckoned in the
Line of Zines.from the Center, fo the Parailel Redius taken and’
/ . meafuredinthe Tangent, dorh give 56 gr. 15 w. for che Rumb from
the Equator: which is the Complement to the former 33 gr. 45 m.
and fo both ways it isfound to be the chird Rumb fromehe Meri-
dian.
But if this Rumb were to be found in the common Sea-chart, ie
Should feemrto be above 47 gr. which is more than the fourth Rumb
from the Meridian. Sy

s+ BytheRumb and beth Latitudes, to find the difference of
Longitude.

As if the places given were A, in the Latitude of sogr, andC

Ne he in the Latitude of 55 gr. and the Rumb che third from the Meri-

ean dian, .

oe | In the Chart, Jee a Meridian be drawn through A, and a Parallel
of Latitude through C, then in A, with A B, make the Angle of the
Rumb from the Meridian BA C, (as was fhewed Cap. 2. Prop. to. )

- Sothe degrees inthe Parallel becween Band C, fhall be found to be
§.gr. 2, the difference of Longitude which was required. Wherefore
the proportion holds for the Seétor,

a | As AB the Radius, eo Mae

As toB C the Tangent of the Rumb fromthe Meridian:
poy So A Bthe proper difference of the Latitudes,

“ | toBC the difference of Longitude.

Accord=

rallel taken from the terms of this difference, and meafured in the Line
of dines, hall thew the diference of Longitude tobe 5 gr. i.

As D Crhe Tangent of che Rumb from the Equator,
to A D'che Radius :

SoC D the proper difference of the Latitudes,
to AD the differencz of Longitude.

According to this, we may beft work by Parallel entrance, firft ta-
king §6 gr. 15 m. for th: Angle of the Rumb from the Equator, out

+ of che greater Tangent, and make it a Parallel Radius: chen rakethe

“proper ‘difterence of Latitudes out of the Line of Meridians, and
carry it Parallel to che former : fo we fhall find itco crofs the Line of
Linesin 5 gr.2. And this is che difference of Longitude required, the
fame as before. ;

But if this difference wereto be found by the common Sea-chare,
it fhould feem to ke only 3 gr..20 m, which ismore thantwo degrees
Jefs than che truth. And yet this error would be greater, if either
the Latitude be greater, or the Rumb fall nearer the Eqiator, as
may appear by comparing the common Sea-charte with the Fable fol-
lowing.

Riz | Theée

A) 2 aa
fe

Nirth and by Weft.
South and by Wf.

La| Long. Diff.

‘|The firft Rum),
from the Meridian.

|La
Gr

——

Onto oan us te 41

| Long
Gr Pap Gr.P

5 . |
ee | FE a ST

North and by Eaft,
Sonth-and by Eajt,

Diff.

Gr- Pur

Or

-| GO

om

Vitae ea

23 90
2475

"26 67

Gr. Par. \Gr.Par,
Ij 10] 61 18

15 41
15 Og
16 26
16 71
br fi Or,
17.05
18 15
18 77
IQ 21
19 73].
29 35
21 07
21 60

23 16

23 07

27 76
28 97)
30 32]
31 84
33°61
35 69
38 24
41 §2
46 15
54 06

Te ee ey MeaOW oa eo Ope BSNL Ret ke oye fe eo A é WSN thed come mcr Py ae

The fecond Rumb Nirth, North-eaft, . North, Nor thene ft,

from the Mcridian. — South. Seuth-eaff, South, South wef.
La| Long. |_Difte tata), £078 Long. | Di/t. | Lal Long. Diff.

Gr Gr. Par. Gr Par. Par.

dese Gr|Gr. Par.\Gr. Par. GriGr.Par. |Gr.Par.|-

ol aE Fa 30] 13 03 a avi 0 ies

| O 42 ra Zul 13°91 33 54. O1l 32 Og! 66.03

O g3{ 2 16 32) F4 C0) 34. 4 02} 32 96] 67 11

124) 3 25 33} 1449] 35 72 63| 33 86) 68 19

1.65; 4°33 34) 15 00} 36 80 04) 34 79] 69 27

2071 § S| 351 Ts sol 37 881 P65] 35 751 7035

2 49} © 49 30) 16 | 38 97 660} 36-75] 71 44

: an GTS? 57, 37) 16 51) 40 05 | 67} 37 80) 72 §2 re

3830 pe 8°66 38] 17 03] 41 13 68! 38 88] 73 60 i

| 3 741 974 70| 17 -§6) 42/27 69} 4000] 74 68
4 16] - 10.82 40| 18 rol 43 | 7OR AAU TO 75177

| 4.59, 1190] 141] 18 65) 44 3 71) 42 43| 76 85 _
5 01] 12 99[| | 42] 19 20] 45 i6| | 2). 43.74} 77 99 :

5 43] '407] | 43] 19 76] 46 54 73| 45 The 7oies a

| gs 85] 15 15 44} 20 a 4.7 A |; 46 57| 80 10

6 28] 16 23 45| 20 92) 48 71 751 4Bti2t) 8rare

6 71| 17 321 | 46} at 50] 4979] 176] 4978] 82 26

7 14, 1840} | 47| 22 11] 50 87] 177] St 5s] 83 344 :
7 58] 19 48 48] +22 721 52 95 73) $3 40] 84 42 ‘
8 91] 2056] | 49] 23 35] 53 03 79] 55 54] 85 52 ;
8 45} 21 Oy | | 50) 23 98) 54121 | 50) $7 82) 86 59 «
8 90] 22.73] 51) 24.63) 55 20 SI; 60 33} 87 67 Me
9 34) 23 | S24 25 30) 5G) 2 2, 63 13{ 88 76 4
9 79| 24 89 3) 25 98) 57 37 3} 66 32] 89 84 i
10 24] 25 .) 154] 26 69] 38 5 | 84] 6999] 90 92 a
10 70] 27 06}. [55] 27 39] 59°53 851 7432] 92 00
Il 16, 28 14 5c 28 32! 6GO°6! 36! 79 631 93 (exe)
II 62] 29 22 57| 28 87] 61 79 871 86 ‘6 94 17
12 08} 30 31 58] 29 64) 62.7 68] 96 10] 95 25
I2 55] 31 39] 59] 39 44] 63 “i Se] 112 57| 96. 3
13 031 32 47 GO! 31 25! 64 94 90 .

FA 2

= = 1
> ie oe

a

|126

NeNSN = 75

- , Sree
09
NN
‘ —
fo)
Qo
(
7 4
>
; NO =
i 6) \
BES a
. ; Si

he third Rune),

, me the Meridian.

North-eaft by ae
South-eaff by South,

—Noerth- bvef bee
_ South-weft by South, ‘f
La| Long. Dif\

79 37).

Lal Long Dift. = Long. Dif.
Ek Ir. Pay \Gr, Par. | Gr Gr. Par\Gr.Par | Gr Greiear: Gr. Par, |
O ; Ole oy | 30 21 03 3608 60 JO'42) 73 16|
4 -.0 66 I 20 Sie hap | So ay. 37 28 | 61 5! 7 7386
ohh aaa, He [52 22 58] 38 49 62] $3 18) 74 50
at 2.00} %3 my 33} 23 38] 39 69 | 63] 34 93] 75, 27;
4| 207; 4 81]. 134} 24 18) 40 89 64] 56 12]. 76 o7
5; 334! Gor] J354 25 Co] 42091 | 65] 57 681.78 17
RG NEEL SN aa fate de) BO)" 25° 82) 49130 66] 5929
“| 4681 8 42 | 37} 20 64) 44 50 67} 69 99} 80 58
B1}:5 13.0) 0 | 38] "27 48] 45 70 | 68| 62 71] 81 78
9g} © 03} To g2 391 28 39] 46 90 69] 64 53] 82 98
cof © 71| 12.03 | | 40] 29 21] 4811] | ol 66 44] 84 19
Pi ay Sb 3i23 41) 30 09). 49 31 | 71} 63 451 85 39
12} 897] 1443} | 42} 3093) 59 51 72) 70 S$] 86 59
r7} 876) Ty O4y | 431 31 88) SI 71 | 731 72771 87 79):
14] 944) 16 84] 144) 32 80) 52024 1 74l 75 ia 89 09
Irs| 10 13] 18 04 |45 33.741 5412 | 75| 77 9231 90 20
16} 11 83] 19 24] 146) 34 60) 5532! | 76} 80 ia 9I 4o
141 12 53} 20 45 " 35907 50 52 77| 83 U5] 92 61
Pirie 23! 2. Gs 48} 36 66) §7 73 78]. 86 25| 6381
rol 12.93) 22 85] | 49} 37 67} 58 93 | 79| 89 Gol 95 01
2.0} 13 64] 24.051 | 50] 38 60} Go ea | 80] 93 27] 96 2
stl 1435] 25 261 4 §1| 3974] 61 33 81) 97 32] 9742
ers 071-26 46" | 2| 4082, 62 54] | 821 1or 85} 08 62
23| 13 80] 37 G6Y | 53] 41 91] 63 74 | 831 106 97] 96 82
4 16 $3| 28 56 | 54). 43 03) 04 94 84] 112 90]101 03
51 17 26] 30074 | 55] 44 19} 60 15 85) 119 90/102 23
6! 18 oo! 31 7 | A 45 37' 07 45 — | 86) 128 451103 43
7| 18.75] 32 47 |s7| 46 58] 68 55 | 87) 139 47
8] 19 50} 33 67 58] 47 82] 6975 | 881 155 Oolro 84
2026 34 3488] [3 28 4911] 7096] { 89, 181 58 197 04.
36 08 50 42] 72 16 |

104. 64). 4

iy peter Silla at Bln Hepa ue I aaa A li ae Ua aaa Re LRG, TA aaa aa
na I hat er ei hea 127
|The forrth Ramby _ North-eaft, Wortherwe j?,
From the Meridian. South-eaft, South- weft.
Lay Long. | Diff. “| Long. | Diff. La, Long, | Diff.
494 lor. Lar, ae Par. ee v7 IGr. ParelGr. Pw Gr|Gr.Par, Gr. Par.
Lie Co} 20) 2p 30] 32 47h ae re Go} 73 46] 84 85|
Ce hg ae _ 100 ele) eat ED 32 C3 43 S4. Of rs 4.9 85:27
2] 52 00];.'2 $3 | 32] 33 81] 45.25 621 79 58] 87 68
“3) 3 OO}. 4 24 [33 34 99! 40 67 63). gt 73} 89 09} |
41 400] 5 GO] | 44} 36 19! 48 07 6: 83 99]. 90 §!
Bee Olt. F 07 351 37 41| 49 50 65, 8631} 9192
6} “6 o01| +8 49 30 35 63! 50 oI [6° 58 73! 93 34
7|~7 02) .9 90] 437] 39 8&8) 52 33 67] 91.23) 94 75}. ,
8}. 8 03] FI 3r | 38] 41 14) 53 74. 68} 93 85] 96 vi
9} 9 04) 1273) 1391 4% 42] 55 15 69} 9658) 97 ¥5
To] Io oy}. 14. 14 | 40 43 71( 56 65 721 99 42} 98 99
PELL ©7115 $6 41} 45 03] §7 98 711 102 43/100 41}.
21 > 4 I6 97 4.2| 46 36] 59 40 72) 105 g8/TOI 82]
Patol 18 3 | 43} 47 72|. 60 er 73 108. 91/103 24] -
14) 14 14| I9 80 44] 49 10; 62 22 74, 1412 43/104 65
Bei cky bZi.sd 2! 45| 5O FO] 63 64 75| 116 i7]106 06
: 16 21] ‘22 63 i ¥f 93] 6F Os Ee 120 17/107 46
17| 17 °35| 2404 47| 53 38] 66 46 77| 124 45{108 89
18! 18 30! 25 45 48| 54 86, 67 $3] |78 129 O8/110 31
19] 19 36] 26 87} | 49} 56 37] 69 29 79| 134. 1OjIIl 72
20] 20 42 28 28 | 2 §7 911 70 71 SO} 139 foll13 14
21} 21 49] 29 70 y1] 59 43] 72 12 Si] 145 65/114 53
22, 22 56, 34 11 §2| 61.09) 73 $44 182 152 4210 ge
23] 23 64] 32 54 53] 02 73! 74 95 | 83 160 10/117 384
4) 24.73| 33 94 54| O4 41! 70 37 84} 168 9 |118 79}.
25} 25 83] 35 35 55; 66 13| 77-78 (3 179 41 120 2)
26! 26 94! 360.77 56167 90! 79 20 Sa@l 192 2rlr2t 62},
27 | 28 06] 38 18 57| 69 71| 80 61 | 87 208 71|123*04
28] 29 18} 39 oa 38) 71 57] 82 02 88) 231 95/124 45
29} .30 32} 41 OI 591 73 49 83 44. Ee 27U 7125 x6
3t 471 42 431 60" 75 491 84 85 90 3

<i, RR eae
ee
% Ip

Secs ee

) Thee CL pa ay Song SS Saran
ie x co DA Me 8 nO Es
t VE : (ed) F eS SES
4
‘

: N28 *

|The fifth ep North eaft and by Eoft North weft and by Weft. p 4
from the Adcridian. — South-eaft and by E: ft. South-weftand by Weft.

a

he bal Long. Diff, ~ La| Long. | Diff. Lal Long, Shiyee
Ne 5 Gr. Par. Gr. Par. Gr Gr, Par. Gr P| Marty Gr. Pips \Grabat .
O -¥O| O 30] 47 10] 54-00 60} 1129 |108 on}
Leo baat im 31! 49 84155 80} }or! 115 9: 1.09 Bal
2) 2 991 3 60 3 Fe 57 €0O 62] Trg IO|LII 6s
3} 449} § 401 133) $2 371 go ‘0 O| 122 34|113 401.
4. Ol: 7 12 341 §4 16] 61 20 64; 125 7OILIS 20
SI 7 5] 89008 25! 55 981 63 00 he 129 181117 CO
6| 9 00| 10 So 36) $7 82] 64 8o 66{ 132 78)118 80
7\, 10 §0| 12 60 | 37| 59 68] 66 60 6, 136 §4|120 60
S| 12 O©) 14 40 38} 61 §7] 68 4o 65} 140 45/122 4
ut 9} 13 §2} 16 20 32 63 48] 70 20 69} 144 53|124 20
a 10{ 15 04] 18 oo 40] 9§ 421 72 00 70; 148 81[1260
‘ [1} 16 56} 19 80 | 4 06 39] 73 80 71| 153 30|127 80
.. [2| 18 og] ‘21 60 +42 69 39] 75 60 72| 158 OO|129 60
Ss 13| 19 62| 23 40 (43 Ol 43| 77 Ao 73|. 163 00/131 40
14] 21 16] 25 20 441 73 48] 79 20 74, 168 26,133 20
r 15} 22°70] 27 00 be 75 58) 81 00 7§| 173 86/135 00
16] 24 26] 28 80 461.77°72\: 82 So 76| 179 84|136 Bol
17} 25 82} 30 60 471 79 89| 84 60 If 186 26/138 60) ©
| 18} 27 39) 32 401 1 48] 82 10) 86 go] 178! 193 171140 gol —
? 19} 28 97] 34. 20 49] 84 36) 88 10 79| 200 69|142 20).
4 20] 30 §5| 3600 50! 86 67] 90 00 80] 208 91/144 00;
i 21} 32 15| 37 80 51| 89 03] 91 80 81] 217 981145 80
es 221 33 70, 39 60 32} 91 43) 93 Go| 1821 228 131147 60
iy 23) 35 381 41 gol 153] 92 88) 95 4o 83] 239 611149 40
a 14| 37 OI] 43 20 541 96 40] 97 20 8 252 85/151:20}
25| 38 66] 45 00 55| 98 98] 99 col’ 185] 268 511153 OO]
261 40 32! 46 8@] J ¥C]101 621100 Bo 861 287 671154 Bo}
27|.42 00] 48 60 yw es 33/102 60 87| 312 361156 60 q
28] 43 67) 50 404 | 5{|t07 12]104 4o | 88) 345 15/158 go]
; iad 45 38] 52 2 | 39}209 98) 106 201 89 406 72\160 20] ~
uae 30. 47 rol 54.00 601112 925108 a “90 de

ae 129
eesti nessa yet ta LEAL EE IS LLL S
The fixih Ramb Eaft North. eaft, Eaft Sonth-caft.
from the Meridian. Weft North- weft, Weft South-weft.

La _ Long. Diff. 5

La _Long. _Dif.
Gr Gr. Par. Gr. Par. Gr. Par. Par

Gr Gr, Par. Par. Gr. Par Gr Gr. Par Par. Gr. Par. Par.

| La Long. _Dift._

“0 fees ol 30 75 98 “75 98 78 39 | éo| 1H 1$2 IS8}L5O 156 78
ay = 2am: 26k. 31 78 78 Br oof: 261 187 071159 40)
Fr 4/831 5°23 32] 81 61] 83 62 62} 192 13/162 OF
Peper 25 7 84] 33 84 48] 86 23 63) 197 36 164. 62!
4) 9 66] 10 45 34| 87 37] 88 84 64| 202 77/197 14)
3} 12 08] 13 06}! [33 99 30] O1 46}. 65 208 38]169 85;
6 14. 5I| 15 68 36) 93 27} 94. 07 66] 214 201172 4
7\ 16 94) 18 29 37| 96 27] 96 68 67} 220 25]175 O
8} x9 37| 20 90 38] 99 31] 99 30 68] 226 57\177 6
9} 21 81] 23 52 3Q9]102 40]101 oI 6g] -233 15|180 3
10] 24 26] 26 13 40/10§$ § 3/104 52 70| 249 96/182 92
TI} 26.71} 28 741 41{108 71/107 14] 471} 247 271185 53
(2) 29 17] 34.368 [42/111 93|109 75 72) 254 90/188 1
{31 31 65| 33 7 43{1T5§ ZO|II2 36 73| 262 93/190 75
14} 34 14] 36 58] | 44lat8 53[114 97] ©] 74) 271 434193 37
13! 36 63) 39 20 45|121 921117 soy | 7§| 280 46/195 9
16] 39 13) 41 81 461125 36|120 20 76] 290 11/198 §9
[7| 41 65| 44 42 471128 87}122 81 77| 300 46/201 21
(8) 44 18 #7 03| 481132 44/125 43 78] 311 621203 82
— {T9! 46 75] 49 6g b | 49/136 O9]128 C4] [ 79) 323 73/206 43
| BO) 49 29] §a 26 s0l139 81/130 65 80] 337 00/209 05
2Ul sr 87] 94 87 [52 143 60/133 27] = Br (351 64/211 66
22) 54.47) 57 49) | 52/147 47]135 88 | 368 00)214. 27
23} 57 08] 60 Io s3'rgr 44/138 46] 831°386 51/216 89
241 59 71] O271{ sf sattss Sojr4t 10] 84} 407 89/219 50
25] 62 36] 65 33 551159 661143 72] | 85] 433 13/222 41
26) 65 04, 57 94 Ei 163 93]146 33 | 4 56)-464 osl224 73)
27| 67 74] 79 55 §7|168 311148 95 [7 -503 881227 3
28] 70 46] 73:17, | 98]172 8o|151 56] | 88) 560 ca]229 95
29} 73 20} 65 78] 159/177 42,154 17 |89 656 08}232 $6
30 75 98! 78 30 601182 18 1§6 78 90 |

o | S

130

The feventh Ramb Eoff nnd by Nurthy. - Eaftteud spsaneba |
from the Meridian. Wit and by North, © Weft and-by. Soathy |

La Long. Ls Diff. La| Long- Dif. 4 Ary bi Long. |. Di ftee de:
Grlor.Par.|Gr var. | || ©? |Gr.Par|Gr. Par i || GriGr.Par. |\Gr.Pard ,
i: ° O ic 158 231053 77h: {| Of 379 351397 SF | -
1} os O2|. 5 12) [341.64 ooli5% od] | OT! 389 56131267;
| 2b yoosl 10 254) 132/269 96}164 02] :| 2) 400 10)317 Bop
3} 15 OS} Fy 38 33175 92/169 .15 | [63] 410 98)322 93
4\: 20 12} 20:50 34/681 95/174. 28 | 4] 422 26/328 of
5} 25 16} 25 63 451188 o41179 go}. 433 94)333 18),
Of. 30 21| 30 75 | 30 194 221184 53 | 446 031338 30);
7)-35 27}. 35 88 37\200 481189 65 458 66)343 43];
81 40 34] 41 03| | 381206 82\104.78 |. 471 80/348 55):
9| 45 42] 46 73 39|213 24|199 90} | 99 485 §21353 68).
iOt yo 52] 51 26 |e 219 761205 03 499 89. 358 Sr j
IT) 55 63} 56 38: 41/226 37\|210 16 | 514. 941363 93).
[2] 60 77| 61 51 421233 a8 a3 287 530 79]369 06},
(3h G5 : 66 63 45 239 90|220 41 $47 $2374 18)
14h 71 o9| 71°76 44/246 84]225 §3 565 221379 31})
1§| 76 28) 76 88} | 45|253 89]230 66 584 03/384 43}.
16) §1 50} 82 OL 401261 051235 79 604 13|389 §6|.
17| 86 7§| 87 14] | 47|268 36|240 91 625 67|394. 691;
184 g2 O21 92.26 481275 80246 04 648 911399 81}.
9} 97 31) 97 39} | 49|283 40|251- 16 | 674 151404 94)
4Ol\192 64,102 51 | 5SO}291 134256 29 | 701 7§\410 06)
Al\108 ae 64. 31]299 O3|261 41 9732 251415 To}!
221113 421112 77 521307 11]266 §4 766 301420 32)!
23/118 87/117 89 53/9159 371278 | | |. 804 86}425 441:
14 124 35|123 021 | 54)323 :82|276 79 |-849 381430 $9)
125 kt29 87|128 14 551332 48]281 92 {901 +4 aaa

20\135 Aq 133 27

i
\29
1310

37}2§0 47/292 17
581359 811207 30
39]369 45/302 431 —
60'379 35 307.551).

146 71143 Jay
id & 441143 en

ine OF 139 40
198 23 158 23

Gr Gr. Par Par Pane

The eighth Rumb of Eaft and Weft, with the Taine anfwering to one dege
i Shae bate and the ate thts to one a of Longitude.

Long. Diff.

NGF Gr. Par.

2 00}

CON AMA wY DL rlol

Oo

Se

WT AaAwp wn

"GO CO 00 CH CO CO CO OO

rer ogee ree ee ee ee ee
~~ ae _

206
rah de
26)
2 26
4 37
2 46
2 56)
2 67
2 79
2.92
307
3 24
3.42
3 63
3 86
4.13
4 44
4 81

O tamie 4
eee erro
_ : ‘ : re

5 °24| 18"

5 76}
6 39
7.18
8 20

9 57.1

II 47 k
14 33).

132 The Ufe of the Meridian Line.
Thefe Tables are calculated for each of the Rumbs.

The firft feven have three Columns, and of them the firft con-
tainech the degrees of Latitude from the Equinoétial to che Pole : the

fecond doth give the difference cf Longitude; and the third che di-
ftance, both of them belonging cothat Rumb and Latitude.

Asinche Table of che third Rumb ; at the Latirude of g0 gr. I
find under the title of Longitude 38 gr. 69 parts, and under che title
of Diftance 60 gr. 13 parts. This fhews chat if ehe courfe held con-
ftantly on the third Rumb from the Equinoétial to the Laticude of
50 gr.the difference of Longitude would be 38 gr. 69 parts Of 100,
_ and the diftance upor the Rumb 60 gr. 13 parts. For hereI reckon the
diftance by degrees, rather than by Leagues or Miles, and fubdivi-

ded each degree into 100 parts, rather than into 60 minutes, forthe —

more eafe in Calculation, and withal co make che Calculation to
agree the becter, boeh with his, and my Crofs-feaff and other Inftru-
ments, i

The ufe of thefe Tables, for the finding of the difterence of Lon-

gitude, is this. Turntethe Table of the Rumb,and there fee what Lon- _

gitude belongeth to either Latitude, then také the one Longitade oug
of the other, she Remainder will bethe difference of Longitude re-
quired. .

As inthe former Example, where the places givenwere A, inthe

_ Latitude of 50 gr. C inthe Latitude of 5 5¢r. and the Rumb the third:
_ from the Meridian: I look into the Table of che third Rumb and

_ and there fiad,

| Latitude 50 gr. Longitude 38 gr. 69 parts.
Laticude §§ gr. Longitude 44 gr. 19-

. Therefore the difference of Longitude ¢ gr. 50.

‘There is another Ufe of thefe Tables, for the defcribing of the

Rumbs both on the Globe, and all fortsof Charts.. For having drawn.
the Circles of Longitude and Latitude, and finding by the Tables, the.
the difference of Longitude belonging to each Rumb and .Laticude :.
If we make a prick in the Chart, at every degree of Latitude, accord-.
ingtothat difference of Longitude, and draw Lines through thofe.
Pricks, foas.they make no Angles, the Lines fo drawn hall be: the-

Rumbs required.~ —
The Wfe of she Eighth Rumb is fomething different from. sik
r

| The Ufe of the Meridian Line. 133

For there being here nochange of Latitude, Ihave fet to each Lati~
tude, the difference of Longitude, belonging to one degree of diftance,
and the diftance belonging to one degree of Longitude.

As if two places fhall be 20 Leagues, or one degree diftant one
fromthe other, inthe Uatitude of s50gr, the difference of Longitude
becweenthem willbe 1 gr. 55 parts. But if chey differ one degree in
Longitude, the diftance oetween them will be only 64 parts, which
fall fhort of 13 Leagues, or at the moft 64.¢7. 28:parts, {uch as 10000
do make a degree. .

6. By the difference of Longitude, Rumb, and one Latitude, to find the:
other Latitude. |

Asif the places given were A, inthe Laticude of 50 gr.C ina grea--

ter Laticude, but unknown, the difference of Longitude 5 gr.3, and the
Rumb che third from the Meridian. : |

--_Inthe Chart ler A B, D.C, Meridians, be drawn through A and C,.
according to the difference of Longitude, one-5 gr. + from the other ;.
anda Parallel of Latieade through A, croffing the Meridian C Din.
D: theninA, with A B,makean Angle of the RumbB A E: fo the
degrees in the Meridian between Dand C, thall be found tobe 5 gr..
the proper difference of Latitude which was required, Wherciore the.
proportion holds for the Secor, |

AsA Dehe Radius,

to D Che Tangent of the Rumb from the Equator ;-
So A D the difference of Longitude,

to DC the proper difference of. Latitude.

According tothis, I take 56 gr. 15 ms. for the Angle of the Rumb-
— from the Equator, out of the greater Tangent, and make ita Parallel:

Radius. Then I reckon g gr. 3 in theLine of Lines from .the Center, |
for the difference of Longitude, So the Parallel taken from che
germs of this difference, and meafured inthe Line of Meridians, fhall.
reach from go gr. the Latitude given, to 55 gr. which is the Lacaude
required.

Or if the Rumb fall nearer co the Meridian, |

\
‘ j \§ »

“ 7. 7 A » ~ = Pelt i ye
4) Cae

rey) Thevfe of the Meridian Line. —

. As BC the Tangent of the Rumb fromthe Meridian, -
isto A Behe Radius: smiy , ager
»- Se BC che difference of Longitude, oe
> 40 A Dthe proper difference of Latitude,::

According to this we may-beft work by Parallel entrance; firtt
take 35 .g7.45 7. for the Angle of the Rumb from the Meridian, out
of the greater Tangent, and make ie a Parallel Radius; then take
5 gr. 2 forthe difference of Longitude out of the Line of Lines, and
carry it Parallel.to the former, ‘till the feet of the Compaffes {tay
in like Points: fothe Line between the Center and the place of this
ftay, being taken and meafured in the Line of Meridians from 50 gr.

forward, hall fhew che Latitude required to be 5 5.gr. as in the former
w + ‘ j

of thethird Rumb, atthe Latitude of 5ogr. I find the Longitude of
38 gr. 69p. Tothisif Iadd ¢ gr. gop.
cude given, the compound Longitude willbe 44 gr. 19 p. and this an-
fwers to che Latitude of 54 ¢r, vA |
Bue if this difference of Latitude were to be found by the com-
“mon Sea-chart, it thould feem to be 8 gr. 13 m. and fothe fecond
Latitude fhould be 58 gr. 13 m. whichis above 3

gre more than the ©
truths :

P

7, By one Latitude, Rumby and diftance, to find

she difference of Lon
gitnde, |

As if the places given were A inthe Latitude of
ter Latitude but unknown, the diftance upon the
tween them, and the Rumb the third from the Meridian
~ IntheChare, let a Meridian AB, and a Parallel A D, be drawn
through A, andin A, with A B, makean Angel B A €, for the Rumb

_ from the Meridian 5 then open the Compaffes
- tude of theplacesto E.F,

ferring them intothe Rumb from A to C, and

len | through C'drawano- -
ther Meridian DC, croffing the Parallel drawn through Ain D, fothe ~
degrees intercepted inthe Parallel from A to D *

» fhall thew the diffe-
» ence of Longitude required to be about 5 &r. 2. Wherefore the pro-
portion holds for the Se@or. | si As

ay.
The like may be found by che Tables of Rumbs, For in the Table

for che difference of Longi-

5°gr. C inagrea-
Remb be6 gr.be-

alles according to the Lati- ]
the quantity of 6 gr, inthe Meridian, tranf-

a en Sane

a

ecole AC the Radius, ue

The-ufe of the Meridien tints. ag

isto A D, equalto BC, the Sine of the Rumb fromthe Meri-
So AC the proper diftanceupon the Rumb, |
. to A Dehe difference of Longitude,

According to this I take the Sine of 33 gr. 45 m. for the Angle of
the Rumb from the Meridian;.and make it a Parallel Radius; then
keeping the Seftor at this Angle, I take 6 gr. forthe diftancey one
of the-Meridian Line, according. to the eftimaced Latitudes of both:
places, and Jay it on both fides of the Seéor fromthe Center: fo
the Parallel taken from che terms of. this. diftance, and meafured in .
she Lines of Lines, fhall fhew the difference of Longitude to be about»

Lite . = es , e . ‘ £ ° “eas
[athis and fome of the Prop. following, where chere is buc one La-
titude known, there may.be fometimes an error of a minute or two, .

in the eftimation of the proper diftance, yes ic may be rectified aca

fecond operation. . & : i RS .
* This Propofition may alfo.be wrought by the Tables of Rumbs, For |
according to the Example, in the Table of che third Rumb, atthe La-
tieude of 50 gr, I find the Longitude of 38 gr. 69 p. and. the diftance -
of 6o\gr. 13 ps to this edd 6 gr. for the. diftance; given; fo the com-
pound diftance will. be 66gr. 13 p. and this. anfwers tothe Longitude —
of 44gr.19 p. then if Ltake the one Longitude out of. the other, the -
difference will be 5.gr. 50 p. #$ before. Ole nv
Bur if this difference were tobe found. bythe common Sea-chart, it.
thould feem ro.be only 3 gr. 20,m.. which is more than.2.gr, lefs than

the cruth. | tig :
8). By. one Latitude, Rami, and difference of Lingitudes, to: find the »
diftance. “ Sis) Bri OG Steins Faw Ye

| Asif the places. given were A, inthe Latitude of sogr. Cina grea-
eer Latitude but unknown, the difference of Longitude between them |
being § gr. 4, and the Rumb the third from the Meridian. |
_ In the Chact lec AB, D.C, Meridians be drawn through A and C, |
according tothe difference of Longitude,, and a, Parallel of Latitude
through A, crofling the Meridian. DC in Ds then in | A. wish.A By.
make an Angle of the, Rumb B A Cy fo the diftance onthe Se
| : | rom

¥3600—tié« HE fe off the Meridian Lint.
from A to Ctaken and meafured in the Meridian, according to the

eftimaced Latitude of the places, fhall be found to be 6 gr. Wherefore ©

che:proporcion holds for che Seéfor.

As AD, equal to B Cy the Sine of the Rumb from the Meridian,
isto A C the Radius: ;
So A Dehe difference of Longitudes,
co AC che proper diftance upon the Rumb:

According tothis, I take the lateral Radius, and make it a Parallel
‘Sine of 33 gr. 45 m. which is here the Angle of the Rumb from the
Meridian ; then I reckon § gr. $ inthe Lines of Lines from the Cen-
ter, for che difference of Longitude: fo the Parallel caken from the
terms of this difference, and meafured inthe Line of Meridians, ac-
cording to the Latitudes of che places, fhall there fhew che diftance
required to be about 6 gr. which are 130 Leagues.

@rif the Rumb fall nearer co the Meridian, chac the lateral Radi-
us cannot be fieted over in his Sine, this Prop. muft be wrought by
iden entrance, and fo alfo ic gives the fame diftance as be-

ore. -

Orwe may find this diftance by the Table of Rumbs. For in the
Table of thethird Rumby atthe Laticude of 50 gr. I find the Longi-
cude of 38 ¢r: Gop. andthe diftanceof Se gr.13 p. To this Longi-
sude here found, I add 5 gr. 50 p. for the difference of Longicude
given: fo the compound Longitude will be 44 gr. 19 p. and this an-
{wers co the diftance of 66 ¢r,15p. Then if I cake che one diftance
out of the other, the remainder will be 6 gr. 2p. for the diftance re-

uired,
But if this diftance were to be meafured onthe common Sea-chart,
it fhould feemto be almoft 10 gr. or at che leaft 197 Leagues, above
77 Leagues more than the truth.

9. By one Latitude, dif-ance, ana difference of Longitude, to find the i:

Rum.

‘As ifthe places given were A, in the Latitude of 50 gr: C ina gres-
ter Latitude,bue unknown, the difference of Longitude between them
being 5 gr-z, and the diftance6 gr. upon she Rumb.

In the Chart [ct A B, D C, Meridians, be drawn through A and C,

and

\

4

The Ufe of the Meridian Line. 137
and a Parallel of Latitude through As then open the Compaffes ac-
cording tothe Latitudes of the places, to E F the quantity of 6 gre
iu the Meridian, and fetting the one foot in A, the other foot hall
crofs the other Meridianin C : and if we draw the right Line AC,
the Angle B AC fhail fhew the inclination of the Rumbro the Meri-
dian, to be about 33 gr. 45 - Wherefore the proportion holds for
the Seéor,

As A C the proper diftance upon the Rum},
isto A Dthe difference of Longitude -
So A C Radius, 3
to AD, equal to BC, the Sine of the Rumb from the Meridian.

According to this, I take the proper diftance 6 gr. out of the Line
of Meridians, and lay ie on both fides of the Secfor from the Cen-
ter; then I take the difference of Longieude § gr. + out of che Line
of Lines, and co it open the Sector in ehe cerms of the former di-
france: fo the Parallel Radius caken from between 99 and 90, and -
meafured in the Sines, doth give about 33 g7- 45 ™; for che Rumb re-

quired.

Bue if this Rumb were to be found by the common Sea-chart, it
fhould feem to beabove 66 gr. and fo almoft the {ixth Rumb trom the
Meridian. |

10. By the Longitude and Latitude of two places, to find their diftance
upon the Rumb.

Letthe Seitor be opened in the Lines of Lines unto a right Angle
(as was fhewed before Cap. 2. Prop.7. ) then cake out the proper dif-
ference of Latitude, and lay iton the one-Line, and she difference of
Longitude, and lay it on the other line, fo as they may borh meet in

-. the Center,marking how far they excend. For che Linetaken from che
terms of their extenfion, and meafured in the Meridian, according to
© sheir Latitudes, thall thew the diftance required.

So if the places given were A and C, Ain the Latitude of §0 gr.
C in the Latitude of 55 gr. the proper difference of Latitude fhali be
the Line A B, and let BC thedifference of Longitude be 5 gr. 3. We
fhall find that A C the diftance upon the Rumb ts about 6 gr. which

make 120 Leagues;
T For

138 0 The U feof the Meridian Line. —

aS ie Sle ere Vi PTE eee ee YS ee YDS ah) * fp tat Ne Pe ee ee ey Bae S Pee aera a or Re ei ee A ee 7 Soe ee hey we
NS eaten S| ea re Fi r PK Fe eR AY eats CaS, ig ge Cana Ae Nite) ry Ph ESS AMON ebm ce bas ees Sa rs rue lt

For inthe Chart, letan occult Meridian be drawn through A, and a. 4
Parallel of Latitude through C, croffing the former Meridian in B,
and aright Line forthe Rumb, from A toC, fo have we a Reétangle,
Triangle A B C, whofe Bafe A C, takec and meafured in the Meridian;
from E below sogr.to F, as much above §§ gr.doch containthe quan-;
tity of 6 gr.

In che tame manner the Sector being opened toa right Angle, inthe
Lines of Lines; if we take the difference of Latitude out of the Line
of Meridians, in his proper place from go gr. to 55 gr. end place it’
on one of the fides from the Center, to refemble AB, shen reckon
che difference of Longitude onthe other Perpendicular Line from
the Center to § gr. 2, inftead of BC, wetfhall have the like Re€tangle
Triangle onthe Seéfor,to that which we had before on the Chart ; and
if we take out the Bafe of ir, and meafure it in the Line of Meridians
from below 5 gr. to as much above §§ gr. we fhall find as before, that ie
containeth about 6gr. or 120 Leagues.

But if this diftance were to be meafured onthe common Sea-chart,
it fhould{eem robe almoft 7 gr, 4,or 245 Leagues ; which is 25 leagues
more than the truth, ! ih

13. Ry the Latitude of two places, and the diftance upon the Ramb, to

find the difference of Longitude. 90-67 11b8P

Let the Scéfor be opened inthe Lines of Lines toaright Angle, then
take outthe proper difference of Latitudes, and lay ie on one of the
Lines from che Center, then take the proper diftance witha pair of
Compafles, and fecting one foot in the rerms of thedifference, turn
the other foot to che other Line of the Seéfor, and it fhall there fhew >
she difference of Longitude required. |

So if the place given were A, inthe Laticude of 50 gr. C inthe La-
titude of 55 gr. with 6 gr. of diftance one from another, we fhall find

their difterence of Longitude tobe abour g gr. 3.

For inthe Chart let a Meridian A B be drawn forthe one,and AC,
A D, Parallels of Latitude for them both: Then open the Compaf-
fes according to the Latitude of the places,to E F che quantity of 6 gr.
inthe Meridians, and fetting one foot in A, having Latitudeof sogr.
gurn the other tothe Parallel of 55 er, and ie fhall there cut off the

required difference of Longitude BC § gr, 3.

o bi n

dj
r..:

The Ufe of the Meridian Line. 2" Bg

Inthe fame manner, the Setter being opened toa right Angle, in che
Lines of Lines : if we take the difference of Longitude our of the Lin
of Meridians inhis proper place from sogr. unto 55 gr. and place it
on one of the Lines fromthe Center; then take 6 gr. che diftance up-
on the Rumb out of che fame Line of Meridians, according to the La-
titudes of the places, and fet the one foot in the cerm of the former
difference, turning the other foot to the other Perpendicular Line, we
fhall find chat ic will crofs ie about § gr.4 from che Center,which is the
difference of Longitude required.

But if this difference of Longitude were to be found by the com-
mon Sea-chart, it would feemto beonly 3 gr. 20m, whicd is more
__than2 gr.10 m. lefschan the truth. |

12. By one Latitude, diffance and difference of Longitudes, to find the dif~
ference of Latitudes.

Let the Sestor be opened in the Line of Lines coaright Angle, and
let the difference of Longitude be reckoned inone of thofe Lines
from the Center; then take the proper diftance with a pair of
Compaffes, and fetting the one foot in che term of the former dif-
ference, turn the other foot to the ocher Line of che Secor, and ie
fhall thence cut off a Line, equal to the proper difference of Latitude
required. i

So if the places given were A and C,A inthe Latitude of so gr.C in
a greater Latitude but unknown, the difference of Longitude becween
chem § gr. z, andthe diftance upon the Rumb6 gr. or 120 Leagues,
we fhall find che difference of Latitude to be § gr.

For in the Chart, let occule Meridians be drawn through A and C,
and a Parallel of Latitude through A, then open the Compaffes ac-
cording tothe eftimated Latitudes of the places co E F the quantity
of 6 gr. inthe Meridian, and fetting the one footin A, turn the other
to the Meridian drawn through C, and it fhall there cut off the Line
D C, which is the difference of Latitude required. |

In the fame manner, the Se&or being opened toa right Anglein -
che Line of Lines, if in the one Line we reckon the difference of
Longitude from the Center to § gr.3, then taking 6gx. for the di-
ftanceourof the Line of Meridians, according to the Latitude of the
places, we fer the one foot in the term of the given difference, and

turn the other foot to the other Perpendicular te we fhall find that
2 it

146 The Ufe of the Mérvidian-line. a
‘itcutsa Linefromit, which takea and meafured in the Line of Meri :
dians, from §0 gr. on forward, doth fhewthe diff erence of Laticude

mon Sea-chart, it would feem to be only 2 gr. 25 m, which is 2 gr.

ve, SGT ee. + wy { ns ae vey a a e « mm A >
a ; ; Pea hs Bs NEPA SUBReM ITED Sy, aetes Ea ¢ rw Oe ee
| ‘RCo Soa

to be asbefores gr. iz een
But if. this difference of Latitude were co be found by the com-

35 m.lefs than the cruch. Such is the difference between both thefe —
Charts. |

THE

ESE ge te IE IS EG LE ALE a a etitetn

,

THIRD BOOK

OF THE

SEC TOR,

Containing the Ule of the particular Lines.

He Lines of Lines, of Superficies, of Solids, of Sines, wich the

laceral Lines of Tangents and Meridians, whereof I have here-
unto fpoken, are thofe which I principally intended, chat little
room en the Seftor which remaineth, may be filled up with fuch parti-
cular Lines, as each one fhall chink convenienc for his purpofe. I bave
made choice of fuch as | chought might be belt prickt on without hin-
? usin the fight of the former, viz. Linesof Quadrature, of Segments,
of Infcribed bodies, of Eguated bodies, and of Metals. |

vo anand

CHAP LT
Of the Lines of Quadratures-

He Lines of Quadrature may bs known by che Jetter Q, and by
fi Gate place between the Lines of Sines. Q_ fignifieth the lide of -
a Square; 5 the fide of a Pentagon with five equal fides, 6 of a Hexa-
gon with fix equal fides, and (0 7, 8,9, and 10. Sftands for the Semi-
diameter ofa Circle, and 90 fora Line equalto 90 gr. in the Circum-.

ference. The ufe of them may be:

5. Tomake a {qaare equal to aC ircle given 3
2, To make a Circle equal tea Square givens

NOUS @

Of the Line of Quadr

BI

Hes

eis D
If the Circle be firft given, take his Semidiameters and to it open
the Seétor in the Points at S$: fo the Parallel taken from between the
Points at Q, fhall be the fide of the Square required, |

If the Square be given,take his fide, and to it open the Seéler, in the
Points at Qs So the Parallel caken from between the Points at S$, fhall
be the Semidiameter of the Circle required, ,

~ Let the Semidiameter of the Circle given be AB, the fide of the
Square equal unto it fhall befoundtobeC D. mee

3. To reduce aCircle Siveny or a Square inte an equal Pentagon, oy — §
other like fided and like angled Figure, | |

Take the fide of the Figure given, and fit it over in his due Points «
fo

; . | Of the Lines of Seg meHhsa. ; 143 a
fo the Parallels taken from between the Poincs of the other Figures,

_ fhall bethe fides of thofe Figures : which being made up with Equal
Angles, thall be all equal one to the other, . 7 |

.

Let the Semidiamerer of the Circle given be A B, the Gide of an
Hexagon equall,to this Circle, thall by thefe means he found to be
GH>5 and che fides of an O@agonto be IK, Other Planes not here
_ fet down, may firft be reduced into aSquare, by che fixth Prop. Su-
perf. and then into a Circle or other of thefe equal Figures, as be-
fore. Wf Pade y ‘vat a:
4. Tofind aright Line, equal to the Circumference of a Circle,or other
part thereof.

Take the Semidiameter of the Circle given, and to it open the Se-.
éisr inthe Points atS; fo the Parallel taken from between the Points at
_ 90 in this Line, fhall be the fourth part of the Circumference: which. -
_, being known, the other parts may be found one by the fecond and.
third Prop. of Lines.

Thus if the Semidiameter of the Circle given be AB, the righe
Line E F fhallbe found to be the fourth pare of che Circumference.
Therefore che double of E F fhall be equal ro the Circumference of
180 gr. and the half ot EF be the Circumference of 45 gr. and foin
the reft.

*

CHAP. Il.
Of the Lines of Segments.

He Lines of Segments which are here pleced between the Lines.

of Sines and Superficies and are numbred by 5, 6, 7, 8,9,
10, do reprefent the Diameter of a Circle, fo divided into a hundred
parts,asthata right Line draw through thefe parts, Perpendicular co
the Diameter, thal! cut the Circle into. two Segment’y.of which the
greater Segment {hall have that proportion to the whole Circle, asthe
partscuthave ro 100. Theufeof them may be,

1, Todivide aCirclegiveninto two Segments according to a Proporticn
given, 2, Lo

544 Of the Limes of Segments:

aed

a. To find 4 Proportion between a Circle and bes Segneente given.

Let the Seétor be opened in the Points of 100 co the Diameter of
the Circle given: fo a Parallel:taken from the Points proportionas
+t ro the great Segment required, fhall give the depth ‘of that greater

egment.

‘Or if che Segments be given, ‘let the Seftor be opened as before;
then take che depth of the greater Segment, and carry it Parallel to the
Diameter s fo the number of Points wherein they ftay, fhall fhew che
proportion ¢0 100. !

. Of the Lines of Inferibed Bodies 146
Asifthe Diamecer of the Circle given were BL, the depth of che
greater Segment L O being 75, doth fhew the proportion of the Seg-
mentO MLN tothe Circle, to be as 75, to 100; viz. three parts of

‘four.
Hence I might thew, if there were any ufe of ir,

2. To find the fide of 4 Square, equal to any known S egment of a Circle.

‘The fide of a Square equal to the whole Circle, may be found by the

former Chap. and chen having the proportion of the Segment to the
Circle, we may diminifh the Square in fuch proportion by chat which
hath been fhewed Lib.t. Cap. 3. Prop. 3.

CHARIS 4
Of the Lines of Iafcribed Bodiese

He Lines of In/cribed Bodies are here placed between the Lines of ©
Lines,and may be known by the leteers D,S,1,C,O;T, of which D
_ fignifieth the fide of a Dadecabedron, I of an I/cofahedron, C of a Cube,
Oof an O&zahedron, and T of a Tetrahedron, all infcribed into the fame
Sphere, whofe Semidiameter is here fignified by the letter S.

_. Theufe of thefe Lines may be, } ie

1. The Semidiameter of a Sphere being given, to find the fides of the five
regular Bodies, which may be infcribed in the [aid Sphere.

2; The fide of any of the five regular Bodies being given, to find the Semsi-
diameter of a Sphere, that will circnm{cribethe [aid Body.

If the Sphere be firft given, take his Semidiameter, and to it open
the Seélor inthe Points at S:if any of the other bodies be firft given,
cake the (ide of it, and fit it over in his due Points : fo the Parallel ta-
ken from between the Points of the other bodies, fhall be the fides of
thofe bodies, and may be infcribed into the fame Sphere.

Dit. schol allie Re Ei y oan a ae

| ieee

146 or | Of the Lines of Metals.

So if che Semidiameter of the Sphere be AC, the fide of che Dode- a
cahedron infcribed fhall be DE, habe ee

Of the Lines of Equated bodies. .
He Linesof Equated bodies, are here placed beeween the Lines of

Lines and Solids, noted with thefe letters D,I, C, S, O, T, of which
D ftands for the fide of a Dodecahedron, I for the lide of an /cofahedron,

_ Cfor the fide of a Cube, Sforthe Diameter of aSphere, Oforthefide

of an Uftabedron, atid T for the fide of a Tetrahedron, all equal one to
the orher. Tise ufe of thefe Lines may be,

1. The Diamatter of a Sphere being given, tofind the fides of tke five ree :
gular bodies, equal to that Sphere. . a
2. The fide of any of the five regular bodies being given, to find the Dia-

meter of a Sphere, and the fides of the other bodies, equal tothe fr ft
body given. |

If the Sphere be firft given, take its Diameter, and toitopenthe .—
SeGor inthe PoinetsatS: if any of the other bodies be firtt given,
cake the fide of it, and fic it over in his due Points, fothe Para'lels ta- md
ken from berween the Points of the other bodies, (hall be the fides of
thofe bodies equal co the firft body given, :

Thus in the lait Diagram, !f the Diamerer of “a Sphere given be BC,

the fide of the Dodecahedron equal co this Sphere,would.be found to be
FG, | ey

CH AP... Ve

Of the Lines of Metals.

He Lines of AZetals are here joyned with thofe before o
~ & lodies, and are noted with thefe Charaers,
of which © ftands for Gold, ¥ for Quickfilver
~vers-@ for Copper, @ for Irony and 1 for Tin.

f Equated
‘OF Q, hyd; Perore ae q
» hfor Lead, Ytor Sil-e
The.ufe of them isto
give: ;

Of the Lines of Metals, 147
give a proportion beween thefe feveral Metals, in their magnitude and
weight, according to the experiments of ALarinus Ghetaldus, in his
book called Promotus eArchimedes.

I. In like bodies of feveral Metals, and equal weight, having the magni -
tude of the one, to find the magnitude of the reff.

Takethe magnitude given ont of the Lines of Solids, and to itopen’
che Seétor in the Points belonging to the Metal given: fo che Parallels
caken from berweenthe Points ot the other Metals, and meafured in
the Lines of Solids, thall give che magnicude of their bodies.

Thus, having Cubes or Spheres of equal weight, but feveral Mee
tals, we hall find, thac if shofeof Tin contain tooco D, the others of
Iron willcontain 9250, thofe of Copper 8222, thofe of Silver 71615
thofe of Lead 6435, thofe full of Quickfilver 5493, and chofe of
Gold 3895. | Be |

2. In like bodies of feveral Adetals and equal magnitude, having the
- sweight of one, to find the weights of the reft.

This Propofition is she converfe of che former, the proportion noe
dire&, but reciprocal, wherefore having two like bodies, take the gi-
ven weight of the one out of the Lines of Solids, and to it open the
Seftor in the Points belonging co the Metal of the other body : fothe
Parallel taken from the Points belonging to the body given, and mea-"
fared inthe Lines of Solids, fhall give the weighs of che body required,

Asif a Cube of Gold weighed 38/. andit were required to know
the weight of a Cube of Lead having equal magnitude. Firft I take
38 /, forthe weight of the golden Cube out of the Lines of Solids,
and put it over in the Points of k belonging to Lead : fo the Parallel
caken from between the Pointsot @ ftanding for Gold, and meafured
inthe Lines of Solids, doth give the weight of the leaden Cube requis
red to be 13 /. :

Thus ifa Sphere of Gold fhall weigh toooo, we fhall find that a
Sphere of the fame Diameter full of Quicklilver fhall weigh 7143, 8
sphere of Lead 6053, a Sphere of Silver $438, a Sphere of Copper
4737,4 Sphere of Iron4.210, and a Sphere of Tin 3895;

U2 3:08

ms | of the Lines of Metalse |

3. A body being given of one Adetal, to make another like wuto st of ane
other Metal, and equal weight. eee

Take out one of the fides of the body giver, and pur it over inthe

Points belonging to his Metal: foche Parallel caken from between the

Points belonging tothe other metal, fhali give che like fide, for the

body required. If ic bean irregular body, fee the other like fides be
found out in the fame manner, )

a Bs

Let the boty given be a Sphere of Lead containing in Magnitude
36 d, whofe Diameter is A,to which Iamto make a Sphereof Iron,of
equal weight: If Irake out the Diameter A, and put it over inthe
Points of & belonging to Lead, the Parallel cakenfrom berweenthe ~
Points of &, ftanding for Iron, fhall be B, the Diameter of the Iron
Sphere required. And this compared with the other Diameter, inthe
Lines of Solids, will be found to be 23 d. in magnitude. 7

4. A body being given of one Metal,to make another like unto it of ancther

Metal, according to aweight given.

Firft, find che fides of alike body of equal weight, then may we ei-
ther augment or diminifh them according to the proportion given, by
that which we fhewed before in the fecond and third Prop. of Solids. —

_ Asif chebody given were aSphere of Lead, whofe Diameter is A,

_and it were required to find the Diameter of a Sphere of Iron, which
fhall weigh three times as muchas the Sphere of Lead: Itake A, and
pueit over inthe Points of b, his Parallel taken from between the.
Points of &, fhall give me Bfor the Diameter of an equal Sphere
of Tron: if this be augmented in fuch proportion as 1 wnto 3,itgiverh
C, for the Diameter required. _ a

CHAP,

\

CHAP. VI." -

Aving fhewed fome ufe of. the

. MLines on the flae fides of the
Secor, there remain only thofe on

the edges. And here one half of the

Of the Lines on the edges of the Settor.

outward edge is divided into inches,
and numbred according to their di-

ftance from the ends of the Sefor, *""

Ass in the Sector of fourteen inches
Jong, where we find 1 and.13, it:
fheweth that divifionto be zinch.
from the nearer end,and 53 inches

from the farther-end of che Seéor,.
The other half containeth a Line

45 4, oucof the number of degrees

this double remainder being added, .
fhall make upthe Tangent of the.
degrees required,

_ As if A Bbeing the Radius,and
BCthe Tangent Line, it were re-

of leffer Tangents, to which the
Gnomon is Radius; They are here:
continued to 75.gr. And if there be
need to produce them farther, Take

required, and double the remain- |
der: fo the Tangent and Secant of :

gr. If wetake 45 gr. outof 75 gr.
the remainder is 30 gr. and the
double 60 gr.whofe Tangent is B D, -
and the Secantis A D: if then we-
add A DtoBD, it maketh BC,the -
Tangent of 75 gr, which was requi-
red. Inlike fort, che Secant ot 62
&r. added to the Tangent of 61 grs

givech ,

quired to find the Tangent of 75:

®

| a
pea SRE eNOS EEE
WA

os

ee \
e ee
.

e
o*?
eee
<
oe

a?
. ance
ee i bos
Raab ES
° reg ; a
? x aa
°
ie *s as > tr
e Ni *S
Bs °
° o9.N *s, H
*. ob y" ‘ af 3
. .* e
O8}..-4 *
se heat
we ae:
* °
a
vi
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aw*
4

= 7 at
rs “de PEEH ie

ae aes feed ae eae — | ie ae oe

a
es
wren”
e
po?
@ posse es
2 et
e
ae?

hay Aes : Vo SpA > SNE ROR LST BSB EI tein Sh eR er, 2p ARUN eer eS aC TO Ve Ae
iy 7a oad Bt igh 4 Aa \
ee (phere y By
-'

er SLO eA.
wy 7 2 ate j Pa P hak ‘ha i)

Neo. we OF the Lines on the edges of the Sector.

giverh che Tangent of 75 eri 30%. and the Secant of 62 gr. added to

the Tangent of 62 gr. givech the Tangeft of 76 gre and fo in the
reft. Theuleof thisLinemaybe,

To obferve the Altitude of the San.

" "Hold the Seétor fo as the Tangent B C,may be Vertical,and the Gao-
‘mon B A, parallel coche Horizon ; thencurnthe Gnomon toward the
Sun, fo chat ite may caft a fhadow uponthe Tangent, and the end of the
fhadow hall thew the Altitude of the Sun. So if the end of che Gno-
monat A, do give a fhadow unto H, it fheweth that the Altitude is 38
gr. 2 if unto D, then 6o gr. and fo inthe reft.

There is another ufe of this Tangent Line, for the drawing of the

hour Lines upon an ordinary Plane, whereof I will fer down thele Pro-
Ppolitions,

1. To draw the hour Lines upon an Horizontal Plane.
2. To draw the hour Linesupon a dirett Fertical Plane.

Firft draw aright Line A C for the Horizon, and the Equator, and
-crofs itat the Point A, about the middle of the Line, wich A B an-

other right Line, which may ferve for the Meridian, and the hour of |

12; thentake out 1g gr. outof the Tangents, and prick them downin
the Equator on both fides from 123 fo the one Point fhall ferve forthe
hourof 11, and the other for che hour of 1. Again, takeout che Tan-
gent of 30 ¢r, and prick it downin the Equator on both fides from 12 :
fo the one of thefe Points fhall ferve for the hour of 10, and che other
for the hour of 2. In like manner may you prick down che Tangent of
45 gr. for the hours of 9 and 3, andthe Tangent of 60 gr. forthe hours
of 8 and 4,and the Tangent of 75 gr.for the hours of 7 and 5.

Or if any pleafe to fet downthe partsof an hour, he may allow
7 gr. 30 m. for every half hour, and 3 &r. 45 m. for every quarter.
This done, you are to cOnfider the Latitude
lity of the Plane. So che Secant of the Latitude thall be the Semidia-

meter ina Vertical Plane, and the Secan

Latitude in an Horizontal Plane.

For example, about London,
Plane be Vertical. If you cake

. it PAV, the Secant of Sf gr. 30 m, out of
the Seéfor, and prick it down i

n the Meridian Line from AtoV, the
Poing

~

I a ee ae eer

of the place, and the qua- _

tof the Complement of che —

|

7

7
SY
Ji
2
-
}

Cen we
ee

ae

the Latitude is Sl ers 30m, andlerthe

yom se ag ~
Wn s* 22 ee

a ————— — ————— =

y

152 ss The Ufe of the leffer Tangent: i

— Point V fhal be the Center: and if you draw right Lines from V un-

0 11, and 10, and the reftof the hour Points, they fhallbe che hour
‘Lines required. |

But if the Plane be Horizontal, thenyou are to take out AH the

Secant of 38 er. 30 m. for the Semidiameter, and prick it downin tke |

Meridian Line from A unto H; fo the right Lines drawn from the

Center H untothe hour. Points, fhall be che hour.Lines required ; on-

ly the hour of 6 is wanting, and that muft always be drawn Parallel co
the Equator, through the Center V in a Vertical, through the Cen-
cer H, inan Horizontal Plane.

This being done, if you fetche Lines A H, H V, to aright Angle
(HAV) the right Line H V the Bafeof this Triangle thall be che
Axis of the ftyle for éither Plane,
3. To draw the hour Lines ona Polar Plane.

4. Te draw the hour Lines on a Meridian Plane.

Ina Polar Plane the Equator may be alfo the fame with the Horizon-
tal Line, and the Hour Points may be pricked on as before,but the hour
Lines maft be drawn Parallel tothe Meridian?

Inthe Meridian Plane, the Equator will cut the Horizontal Line

with an Angle equal tothe Complement of the Latitude of the places
then may you make choice of the Point A, and there crofsthe Equator
‘with aright Line, which may ferve for the hour of 6: fo the Tangent
of 15 gr. being pricked down in the Equator on both fides from 6, fhall
ferve for the hours of and 7; and the Tangent of 30 gr. for the hours
of 8and4, andthe Tangent of 45 gr. for the hours of 3 and gs and
the Tangent of 60 gr. for the hours of 2 and 10; and the Tangent of
75 gr. for the hours of 1. and 11. Andsf you draw rightLines through
thefe hour Points, croffing the Equator at right Angles, they thall be
the hour Line required. |

The Subftilar will be the fime with the Hour of 12 in the Polar

‘Plane, and with ehe hour of. 6 inthe Meridian Plane: the Axes of the
ftile may be Parallel to che Suitilar in either Plane according to the
diftanceof the third hour fram che Subftilar. :

§. Todraw the biar Lines ia a Vertical Declining Plane.

Firft, draw A V the Meridian, and A E the Horizontal Line,
a crofing

The Ufe of the belfer Tangent. 353

&

AS A CM A ON TS GA oD. . ER

croffing one the other at right Angles in the point A.
_ 2.Then take oat A V,the Secant of the Latitude of che place,which
you may {uppofe to be § f. gr.30 m, and prick it down in the Meridian
Line from A unto V: ees
3-Becaufe ic is adeclining Plane, and you cen fuppofe itte decline
42 fo

354 The vfeof theleffer Tangent = =
4.0gr. Eaftward, you areto wake an Angle of the Declination upon
‘the Center A, below the Horizontal Line, and tothe left hand of the —
Meridian Line, becaufe the declination. is Eaftward, for ocherwifeic _

a oe VPs) ee -- fe ay ve ve. Mie 1% 4 Peat aka
\ eee

thould have been to the right hand, if the Declination had been Welt-
ward. oe |
4. Take AH, the Secant of the Complement of the Latitude out of
the Sector, and prick is down inthe Line of Declination from A unto
H, 3s you did before for che Semidiameter in the Horizontal Plane.

5. Draw a Lineat full lengeh through the Point A, which muft be
Perpendicular unto A H, and cut the Horizontal Line according tothe
Angeles of Declination, and ic will be asthe Equator in the Horizon-

‘tal Plane. | : “a

6e Takethe hour Points out of the Tangent Line inthe Seéfor and
prick them dowa in this Equator on both fides from the hour of 52
arA. f

7. Lay your Ruler, and draw right Lines through the Center H,and
each of thefe hour Points: fo have you all the hour Lines of an Hort-
zontal Plane, only the hour of 6 is wanting, and that may be drawn
through H Perpendicular to H A. | |

Laftly, you areto obferve and mark the Interfections, which thefe
hours lines d» make with A Eche Horizontal Line of the Plane: and ~
then if you draw right Lines through the Center V;, and each of thefe
Interfeétions, they fhall be che hour Lines required.

The LineH F drawup tothe Horizon, and Parallel tothe Meridian,
will give the Subftilar VF: The Line FG drawn Perpendicular ta-=
V F, and equal co F H, will give VG, the Axis of the file, M

j ‘
el

%

6, To prickdown the bosr Points another way.

Having drawn a right Line for the Equator as before, and made ©
choice of the Point A, for the hour of rz: you may at pleafare cut
off two equal Lines A ie,and A 2. Tien upon the diftance between |
roand 2, make an Equilateral Triangle, and you fhallhave Bforthe —

— Center of your Equator, and the Line A B fhall give the diftance from _

Ato 9, and from A to 3.. That done, take out the diftance between _
aod3, and chis fhall give che diftance from B unto 8, and from B
unto 43 again, from 4tort,and from 8 unto'r, andalfo from 8 to 7.

So have you the hour Points, and if' youtake our the diftance B1,B 3.
Bs, @c. You may find she Points not only for. the half hours, bucalfo ~

tor the quarters,
?

But

bi

| _ The vfe of the leffer Tangent. 155
But if it fo fall our, that ome of chefe hour Points fall out of your
Plane, you may help your felf by the larger Tangent, both in the Ver-
tical, and Horizontal Planes. : |
For if at the hour Points of 3 and 9,in the Scheme of the Horizontal

*

and Vertical Dials, you draw occult Lines Parallel to the Meridian ;

che diftances D C between the hour Line of 6, and the hour Points of
3 and 9, will be equal to the Semidiameter A V ina Vertical, and AH

in a Horizontal Plane, and if they be divided in fuch fort as the Line”
A Cis divided, you fhall have the Points of 4, and 5, and 7, and 8,with

their halfs and quarters, .

Asinthe Horizontal Plane, take out the Semidiamerer AH, and
make ita Parallel Radius by fitting it over in the Sines of 90 and 90:
Then take 15 gr. outof she Jarger Tangent and lay them on the Lines
of Sines, where they will reach from the Center unto the Sines of

15 gr. 32 ms. therefore take out the Parallel Sine of 1s gr. 32 m, and it

~

fhall give the diftance from 6 unto 5, and from 6 unto 7, inyour Ho-
rizontal Plane. That done, take ous 30 gr. ourof the larger Tangent,
and lay them on the Sines, from the Center unto the Sines of 3§ gr.

“36 m. and the ParallelSine of 35 gr- 16m. fhall give you the dittance

trom 6 unto 4, and from 6 unto 8, in your Horizontal Plane.. The like

may be done for the half hours and quarters.
~ So alfo in the Vertical declining Plane. If you firft take outthe Se-

| ~ cant of the declination of the Plane, and prick it down in the Hort-

zontal Line from A unto E, and through E draw right Lines Parallel
to the Meridian, which will cutthe former hour Lines of 3 and 9, or.
one of them in the Point C; then take out the Semidiameter A V, and
and prick i¢ down in thefe Parallels from C unto D, and draw-right
Lines from A untoC, and from V unto D, the Line V D fhall beche
hour of 6, andif you divide thefe Lines A Cand DC, infuch fore as
you divided the like Line D Cin the Horizontal Plane, you fhall have

_ allghe hour Points required.

r you may find the Point D, in the hour of 6, without knowledge
either of Hor C. For having prickt down A V in che Meridian Line,

‘and A E inthe Horizontal Line, and drawn Parallels to the Meridian

through the Points at E,you may take the Tangent of the Laticude our
of the Sector, and fit it over inthe Sinesof go and go: fo the Parallel
Sine of the Declination meafured inthe fame Tangent Line, fhall there
fhew the Complement of the Angle D V A, which the hour Line of 6

‘maketh with the Meridian; then having the Point D, take our the Se

x2 midiameter-

ee
" Val }
i aa

yt

156 ae The Vfe of the leffer Tangent.

midiameter V A, and prick ic down in thofe Parallels from D unto C:: q

fo fhall youhsve the Lines DC and A Cro be divided as before,

The like might be ufed for the hour Lines upon all other Planes, But
I mut not wrice all chat may be done by the Seétor, It may fuffice thae
T have wrote fomething of the life of each Line, and thereby given the
ingenuous Reader occafion to think of more.

The Conclufion to the Reader.

| hide wel known to many of yor, that this Seor was thus contrived, the
moft part of this Book. written in Latine, many Coptes tranfcribed and:

difperjed more than fixteen years fince. I am at the left contented to give way -—

that it come forth in Englifh. Not that I think it worthy either of my labour,
or the publick view, Lut partly to fatesfie their importunity, who not under-
Standing the Latine, yet were at the charge to bay the Inftrument, and partly.
for my owneafe. For as it ts painful for ethers to tranferibe my Copy, fo it as.
sronblefome for me to give (atisfattion herein to all that de/ire it. If J find.
ats £0 give yor content, it (hall inconrage mse te do the like for my Crofje
ftaft, and fome other Fxftruments. In the mean time bear with the Printers

faults, and fol reft,

Grefham Coll. 1Maij. 1623. E.G.

.
FINIS.. |

es

‘THE

er ea ee PS ee eS AORN iy? me BR ey i t Ape 5 =
‘i a ama E r! ea B
(eat ) 5 4 ~
a) Ps . s aes
‘.
; “ : F
_ . ‘ A a. i?
x i '
ql ’
\
i _*
-
;

ALTERED:
AND

Other SCALES

ADDED: :
With ahs and Ule thereof.

ens:

Invented and ‘written by. ah Samuel: —

Fofter, fometime publick Profe(for of Aftronomy, |
a Grefham Colledge in London.

And now Publithed by VV, L.

saeieneee Gemtrentesinnss anemee

LONDON, .-

Pringed by Andrew Clark, 1673.

te eee

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BERRA RSA SOARED SE: ORAR BARRE RARER SD IRS
| BINNS e ee eeeTee sueeereveveereiess
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ALTERED.

a
3

| CHAP. L
Of the Sector in general,

Sep Monet the many Writers that bave been upon the
Sector, Mr. Gunter hath done belt, the Lines of his In-
ftrument being moft in number, ard of che mot for-

partly becaufe he had no Line of verfed Sines, (of which inhis Book -
there is good Ufe, and might have been much more ) but inftead
thereof he is compelled to ufe che Line of right Sines, which is bur
half of the whole Scale of verfed Sines, and belides the pares of it ftand
the contrary way, fo that the fitting of the proportional terms where=.
by to work with halfe the Scale inftead of the whole, and then the
application of the parts, from one endo the other, will be nora little

troublefome. :
) To.

“160 | My, Samuel Fofter his a
_ Toremedy thefe and other like defects, I have altered the formo ;
the Se&tor once more, VRS | : a

t, By diminifhing the number ‘of the old Scales, for inftead of
two of each kind, here is but one. . a

_ 2. By taking the Meridian Line quite away ; and fupplying the Ufe

thereof by other means.

3. By bringing the Scales of Tangents and Secants ro the Center.

4. By addinga Lineof che Verfed Sines, and fome other Scales of
g0od ufe, He

5. And by changing the form of working upon the Inftrument : of
all which things I fhall give an account in this following Treatife : but
fir ic will be requifite to defcribe the order and difpolition of the
Lines, how each of themistobeplaced.

CHAP, I.
How the feveral Lines are difpofed upors the Settor,

XX 7 Hereas in other Sectors there are always two Lines of one :
. kind, upon each Leg one, anfweringtothe likeScaleuponthe —__
other Leg, in this there is bue one Line of one kind, from whence it 4
comes to pafs, that one fide or flat of chis Sector holds all the Scales |
that are drawn from the Center, and do fill up both fides of the other,
_ and by thismeansthe other fide is free for other Scales.
- Upon one Leg therefore of the firft fide are:
I. A Line of equal parts.
| 2. A Line of Solids, and between thefe two Scales and the edge,
there are inferted two particular Scales more. Namely, I,
3. Of Infcribed bodies. } i
« 4 Of Squated bodies,with a Scale of Metals. So again, upon the
-other Leg, there are:
5. The Lines of Sine:. |
6. The Lines of Superficies, and between them two Scales and the
-edgeare infcribed twoother particular Scales sas,
7» The Line of Quadrature. And,
8. Segments. Allthefe Scalesare drawn from the Center, andbe-
ing meafured from thence, are all of one length: and do lie at {uch —

Angles

alteration of the Sectors 161
Anples one fromanother, and to the edges of the Sector, will sive
them convenient diftance. So that this one fide of the Inftrument dah
~ now contain fo many Lines of Scales, coming from the Center, as were —
before on both fides. | Tighe ee

Upon the fecond fide of the Seétor are four Scales, two upon one
Leg, and two upon the other. As namely upon one Leg,

9. Verfed Sines, with a Zodiack Line annexed to tt.

10. A Line of Tangents going up to 63. gr. 26 mm.

11. A Line of Secants going up to60gr.:

12. A Line of Chords going up to go gr.

All thefe are drawn from the Center, and all of one length with
chofe on the other fide of the Inftrument. The Radius of the Verfed
Sines, Tangents, and Secant Lines are jufthalf of the whole inicribed
Lines, and fowill be of very good ufe inthe working of proportions,
and in the projecting of the Sphere very commodious.

The defcriptions of each Scale may be made by thofe Tables, and In
ehat manner that Mr. Gwater hath directed. ,

Between thefe four Scales may be placed other Scales of good ufe,
‘tending cowards ( though not ruaning upto ) che Center,as a Tangent
of three hours of good ufe in Dialling, and other the like Lines.

Of the other Lines infcribed on the edges and [pare places of the Seétor.

If the Seétor be made of wood, ic will require fome comperent
thicknefs, fo ehat the edges will be large enough to receive fome ufe-
ful Scales alfo, - |

The Seétor then being opened, and fo made a ftreight Rular ; the
outer edge hath infcribed upon it the three ufual Scales of Logarithme-
tical! Numbers, Sines and Tangents. The inner edge hath two Scales
uponeach Leg, one pair of chofe Scales upon one Leg,is to find the
mean Diameter, andone of them is divided into 14. equal parts, the
other (of the fame length with it ) is divided into 26 equal parts, each
of them fubdivided decimally. The other pair of Scales upon the
other Leg is alfo divided equally, one of themcontaining four parts,
which are to reprefent feet, and the other being of the fame length
‘¢ divided into 400 parts, reprefenting Inches of the former
Feet, and each of chefe reprefentative, both feet and inches are {ub-
divided decimally. And again, upon the ewo flat edges of the
Seétor thus opened ( near the outer edge ) are infcribed two peculiar
Scales (upon one edge ) of equal parts for it Sas Ale pene |

Jpon

Y at wn ee) ae RD eS a ee a 5

162 © _ Mr. Samuel Fofter bis |
Upon the other flat fide are two Scales more, each equal to the other,
both of a juft foorinlength ; one is divided into 12 inches, and each
inch fudivided Decimally, the other is divided into 10 equal parts,and
each of them again iato 10. Thefe two Scales ferve for trueinch and _
trve foor meafure. c |
In this manner are the Lines difpofed, now follows,

CHAP. Iti,
The general ufe of the Sector ,and the manner of working upon its

Wee works that are performed upon the other Se€tor when it is
fhut, are alfo performed by this, and in the very fame manner.
Bucthe chief ufe of the Seétor is, by having three terms givento

find a fourth proportional chat the fourth may be to the third as the {e-
condisto the firft. And if thefecond and third cerms fall outtobe
the fame, chenche proportion is called continual, becaufe the fecond
term is twice repeated, and fo the next term continued in the fame pro-
portion toit, that the fecond was co the firft. But if the fecond and
third terms be different, then isthe proportion called difcontinual,.
_ becaufe the proportion that is between the firft and the fecond, though

it bemade good again between the third and che fourth, yerie dif-
continued between the fecond and the third terms. Now becaufe this
kind is molt frequent, and the former may be referred to this, ( if the
fecond term being twice repeated, be takenas two, namely, asthe fe-
cond and. third : } I will thew in general

How by three terms given in any kind, tofind a fourth.
Firft, Difpofe the three terms given fo, as that when they are of

divers kinds, the firft and che third may be of one kind,and thefecond ~ i

and fourthof another, though this difpofition be not always necefla-
ry, yet for che working upon the Seétor it will for the moft part be.
convenica:.

When the terms are fo ordered there will three things hence fol=
Jow. The firft will be to know upon what Scale the Work wil! be per="
formed, when the terms are not all of one kind, the other «wo will be
Rolesand Direétions in what manner to work. |

Firit, Therefore you muft refer eachof the two firttermstoits

proper

\

; si CF: TES. : } Se
4
& 1
. 163

3 alteration of the Setter,
proper Scale, then comparing the fecond cerm with the firft, fee whicta
of themis greateft: For upon that Scale co which the longeft of che
two terms befongeth, mutt the whole work be performed. Then the
two Rules for che manner of working are thefe.
| 1. If che fecond term be lefs than che firft, you muft then count che
_ firft and the chird terms (being both of one kind ) laterally upon cheie
proper Scale, and the fecond term being taken out of his proper Scale,
and put over parallelly inthe term of che firft, thall open the Leg of
the Sector fo, as che fourth term may be taken parallelly over from the
term of the third:and being fo caken,it muft be meafured upon the Scale
fron whence the fecond term wastaken, and (0 it fhall receive its juft
value.
2, Butif the fecond term be greater chan the firft; you muft chen
count the fecond laterally, and in thecerm of ie pue over che firft pa-
rallelly, each being taken in his proper Scale, and this work shall
open the Legs of che Sector fo, as chat che third term being taken oue
of the fame Scale wich the firft, and entred parallelly, fhall May in
that point of the Scale on which the {econd term was counced,and will
givethe quantity of the fourth term required.

The manner of working then in general according to thefe twoRules willbe this :

| Inthe firft cafe; where the fecond term is lefs than thefirft, lee AD
> be the firft cerm, BD G the fecond, and A B the third. |

| Count A Dethefirftcerm, upon his proper Scale, then with your
_ Compaffes take the fecond term,

| which we fuppofetobe DG, ©§ De—————__—A
and fetting one foot of that ex- e ‘D
centin D, the end of the firft, Bt-——A
ecurn the other foot about, and Fr—B

openor fhue the Sector, tillthe
foot being turned about in the
Ark EG, do only touch fome
one Line in the other Leg of the
_ Seétor, neither going beyond tt,
nor fhort of it, as here it doth
at G, fois the Seftor opened toa
true Angle for this Work.

Y¥ 2 Again,

Wo oh BO de Ta Me \ PEE SEALE SEBS Oo Oma Maer NAS cera oe

atte
(
@

164 Mr. Samuel Fofter has |

Again, uponthe fame Scale A D, whereonthe firft term was num-
bred, count che third cerm ABs, and fJaftly from B, the exeremity
thereat, take che leaft diftance to the Scale A G, as liere is exprefled
by BE. So fhall B EB be che length of the fourch term, this Line. BF
therefore being meafured upon the fame Scale from whence the fe-

cond rerm DG was taken, fhall give che quanticy of the fourth term
required. | |

Or if AB had been the firft term, and BF the {econd, then the .

Sector muft have been opened by putting over BF from the rerm B
C cill the Ark or Motion of the foot of che Compaffes, EF, had on-
ly touched che Line A F.) And when the Legs of the Seftor are fo opes
ned, count A D thetbirdcerm upon the fame Scale whereon the firit
term AB was counted, and from the extremity of ir, atD, take the
leaft diftance. from the fame Leg A F, which here willbe D G, fo
fhall D G ( being meafured upon the fame Scale of the fecond term )
give the quantity of the fourth term required.

_ Inthe fecond cafe, when the fecond term is greater than the firft.
Suppofe D G be the firft, and A D the fecond, BEthethird, take the
firftcerm DG, out of bis proper Scale, and count AD the fecond

in bis proper Scale,. then from Dthe extremity of the fecond term, -

open the Seétor, making DG ( when it is‘turned about in EG ) only
to couch any one Scaleinthe other Leg of the Sector, as AG, when
the Sector is thus opened, take BF che third term, out of the fame
Scale from whence the firft term DG was taken ( which is his pro-
per Scale ) and keeping one foot of che extent always upon the Line

A D, remove it to or from the Center A,. till it ftand infome point of

the Line A D, foasthe other foot being turned about inthe Ark EF,

may juftly couch the Line A G, uponthe other Leg, and when you aw

have fo fitted itexaatly, obferve the Pointin A D, in which the foot
of the Compals refteth, which fuppofe to be the Point B. So fhall A B
givethe quantity of the fourth term required.

Or if B EF had been the firft, A B the fecond, and D Gthethird,
then moft the firft B F, taken out of its proper Scale, have been fer
upon 5 the extremity of the fecond AB, and by irthe Line A FG
muft have been opened, and this being done, the third term D G be-
ing taken from the fame Scale ( from whence the firftterm BF was
taken ) and fitted intill one foot of it ftanding uponthe Scale A B, the

other being curned about inthe Ark EG, will only touch the Line —

A FinG, fo fhallthePointD, wherein then it ftands, give che quan-
eityor A D the fourch cero required. This

"ey OO a oe enn Peta, te ey SF Pig) whe A ya . i? a
Lat pipe DE Aa By rac? bie Ss ia ie AN eats 5 n EyATYER > Tay eM

alteration of the Settor. 165"
This may ferve for a general view of the manner of working upon.

thefe fingle Scales, and how one of a kind may ferve to perform any

sity in this Seétor, as well as cwo-have-done formerly in other
eCtors. | }

As alfo here may be feenthe manne¥ of Lateral and Parallel en-
trance, and finding known and unknown quantities: Ir may Itkewif
be here known what is meant by thefe Phra(es. :

1. Opening the Seftor to any Line, length, or diftance, namely, to
open or fhut any two Sceles upon feveral Legs of the Sector, till one
foot of that length being fet in fome Point of one Scale, the other
foot when it is turned about, may only couch the other Scale, foas noe
to go beyond it, nor fall fhort of it. | 3

2. By taking any Line, length, or diftance, namely, from fome.
Poincin one of the two opened Scales, to take the leaft fromthe other
+ Scale. |
| 3. Entring any length or diftance, namely, to carry one foot of a
Jength taken in your Compafs upon one Seale (from or towards the
Center ) till che other being turned about, may jaftly couch the other.
Scale. Tnefe terms are ufed to avoid needlefs circumlocution.

Tc may farther alfo be obferved, that this way of working is more
fpeedy than that upon other Seétors, asby a little practice will quick-
| ly betfound. | oe
| And laftly, the truth of thework will eafily appear, if it be confi- _
'. dered that in every work thus performed, A B Fand AD Garetwo-
like Re@tangles 5 as inthe other Seftor the work went upon two like |
_ Eguicroral Triangles, in bot therefore the ground of the work is.
| alike‘ good, both being grounded upon the fimilitude of two plain
| Triangles. , ’

| Now tethis general direttion for working, I have added examples in feve-
| yalkinds, whereby the Rules before given may the better be ander ftood,
| “and what 2Ar-Guorer andothers have publifhed in their Books, msay the
snore eafily be applied to ths Inftrument. ag

| | a, CHAPy.

ie i

166 _ Mr Samuel Fofter bis |

Cc H A P, TV,
Examples in feveral kinds.

1. Three numbers §25 39,44, being given, to find a foarth proportional,

Cite is wrought upon the Line of equal parts, and becaufe the firtt
number is greater than the fecond, therefore I count the firft
number 52, upon che Line of equal parts, and fromthe fame Line I
eakethe fecond term 39, and fet ie upon $2, and turning the ocher
footabout, I open the other Leg of che Se@or, till ehe fame footdo
juftly couch fome one Line on the other leg of che Se€tor which iffueth
from che Center,neicher going beyond it, nor falling fhort of it,fo are
thofe two Scales opened fitly to perform the work, then I count the
third term of 44 upon the faid Line of Lines, and from the end of ie
to the fame Scale on the other Leg, I take che leaft diftance, this be-
ing meafured inthe Scale of Lines, giveth 18 for the fourth term ; fo
that as §2 isto 39, fo 44 to 33,
But if the given numbers had ftood thus, As 24.to §2, fo18to -

what? Here becaufe the fecond number is greater than che firft, I
take 24. out of the Line of Lines, and fet one foot of itin $2, counted

-uponthe fameLine, and J openor fhuc the other Leg of the Seétor,

till che other Foot being turned about, will only couch fome one Line
onthe other Leg of che Sector which iffueth from the Center - When
che Sector is thus opened, I take the third number 12, out of the
Line of Lines, and keeping one foot always upon the Line of Lines,
I remove it fo long till che other foot being turned about, will only
touch the former Line on che other Leg: Then fhall I find it to ftay
npon the Line of Lines, atthe number 39, which is’ the fourth pro-
portional. .

Inthe fame manner all proportions in numbers may be wroughe by
the Lines of Solids andSuperficies. But if youhad three Lines given,

and were to find a fourth proportional Line, you muft then work upon

the Line of Lines only.

2e Three

ye

alteration of the Sector. 367
2. Three Sines being given, to find.a fonrth proportional Sine.

His is to be wrought upon the Line of Sines only, Let the Sines
\B given be 90, 30, and 23 3 ; here becaufethe firft rerm is greater ;
therefore I muft work bythe firft Rule; and forthe foureh term being
taken and meafured upon che Line of Sines, will be the Sine of 11 4 gr.
required. :

Bur if the Sines were of 36,72, 18 gr.. then work by the fecond
Rule, becanfe the fecond term is greater than the firft, fo thall your
Compaffes ftand at che laft of your work, at the Sine of 30gr. Or
becaufe all the four terms are of one kind, you may change the places
of thefecond and third, thus: 36, 18, 72, and fo working according- -
ly by the firft Rule (becaufe the firft cerm is greater chan che fecond )

you fhall find the fourth proportional Sine to be 30 gr. as before.

In this manner you muff work when all the four terms are of one kind, and :
fo wrought upon one Scale alone. But if the terms be of feveral forts,
then muft two of the four-termes be taken froms one Scale, and two frons ;
amther, As inthe examples following will appear. |

3: As the Sine of GOgr. tothe number 35, [othe Sine of 48 to what :
atmber 2 |

“N folutions of chiskind (becaufe the firft and fecond, and alfo:the -
third and fourth, are counted upon feveral Scales, as here che firft :

and ebird are taken upon she Scale of Sines, and the fecond and fourth .

are taken upon the Scale of equal parts:) You muft firft trie whichis .
preaceft of the firft or fecond cerms.. As here take the fecond term
35, out of the Scale of equal parts, and meafure it upon the firftrerm ;
of che Sine of 69. Now becaufe the Sine of 60 is greater, therefore

| the lareral work muft be done upon the Scale of ‘Sines, and che fe- .
‘cond and fourth rerms mutt be taken in your Compaffes from the Scale

of equal parts, which is their proper Scale; wherefore in this ex- .
ample take 35 out of the Scale of equal parts, and with one foot of -
that length fet in the Sine of 60, open the other Leg till chat extent -

| will jufttouch fome one Line on the other Leg of the Setor which .
| iffueth from the Center, the Seétor being fo opened, take fromthe .
- Gine of 48-the leaftdiftance,.to the former Line on the other Leg,

this .

Y Any \

us et, LS ht rat yt NE Md 7 0?
CG AEN 9 Cee NR BRRUNT he SCID oad tn Sean 7 cc
beh UA, ao a SP Bt eee | 4
+ CARY ;
H

168 Mr, Samuel Fofter zs a
‘this diftance meafured upon the Scale of equal parts, fhall give 30 the ©
number required : Therefore as 60 g7.t0 35,fo 48 gretoZ00

But if it had beenas the Sine 60 is tothe number 90; {othe Sineof —

48 to whac? Here if youmeafure the number go with the Sine of 60,
you fhall find che number 90 to be the Iongeft extent. So thatnow the -
lateral work muft be uponthe Line of Lines; I rake therefore the Sine —
of 60 out of the Sines, and fetting one foot of chat extent uponthe —
che number.go in the equal parts, with che other foot turned about, _
I open the other Leg, till I fee che fame foot only to touch fome one
Line on the other Leg of che Sector, which iflueth from che Center,

Note that what Line foever I begin to work with, I nuft be fare always to
continue and end with the (ame, but that Line onthe other Leg, which
lieth next the inner edge of the Se€kor, always woof? convenient.

Then againI take the Sine of 48,and keeping one foot of that extent —
continually upon the Line of Lines, I remove che fame till I find che
other foot juftly to touch the former Line on the other Leg 5 and then
I fee the other foot to ftay upon che Line of Lines onthe number 77 $5 _
which ts the number fought.

4, As the Sine of 60, % to the Tangent of §5 gre So the Sine of 50 to the
Tangent of what ark? |

ZT irft, to know upon what Line to work, I take the Tangent of
| 55 gr. and fet te coche Sine of Go gr. and becaufe I fee the Sineof
6o0%0 bethe greater, I knowchat the work muft be done upon the Line
of Sines. And by the firft Rule accordingly cherefore I take the Tan-
gentof 55 gr. and from the Sine of 60 I open the Se€or to fome one
Line on the other Leg of the Seétor, which iffuech from the Center
according tothatexient ; then I take the leaft diftance from che Sine
of sotothe former Line onthe other Leg, and meafuring it upon the”
Tangent, I find itto reachto the Tangent of 51 4, which is the Tane
gent required,
But if the terms were asthe Sineof 40 isto the Tangent of 55 gre
fo is the Sine of 50 to what? Then meafuring the Tangent of 55 gr.
upon the Line of Sines, and finding the firft term 40 to be lefer, 1
fee that the work muft be done upon the Line of Tangents: Where-
fore I take the firft cerm the Sine of 40, and fetting one foot of chat

extent

:

Ce da ole ape aR aS at ae
,* of bs ‘
A 4p! “
al? : ion’

alteration of the Sector. poe

extent upon the Tangent of 5%. by turning the other foot of the Com-

paffes, T open the other Leg of the Seétor, eill the other foor do juft-
| Ty couch fome one Line on the other Leg of the Sector, which ifluerh -
- from the Center; then] take the fecond term, che Sine of 59 gr: and
fetting one foot of that extent uponthe Scale of Tangents, uncill the
other foot being turned about, will juftly touch che former Line on
the other Leg: I find the Compaffes to ftay upon the Tangent of
§9 agr. which is the Tangent required. Seay
The like may be done upon the-Sines and Secants, or Tangents or
Secants, when any fuch queftion fhall berequired. And the like may
be done in Tangents and Numbers, or equal parts, by che joyne ule:
of chefe two Scales, which is frequent in Menfurations of upright.
bnildings, | Sais

5. Having three numbers, to find a fourth in duplicated proportion. -

| ag His work is performed by the two Scales of Superficies and Lines
joyntly ufed. Letthe example be as 22 (0.24. : fo 64.to what num-
ber ina duplicated proportion ? Here the two firft terms are of one
kind, and the two latter will therefore be of onekind. Wherefore
to know upon what Scaleto work, it willbe beft to change the places
of thefecond and third cerms, that fo the firftand third may be of one
kind, as alfothe fecond and fourth. Thusas 32 to 64, fo 240 what?
Now in this difpofition of terms you muft firft meafure 32 ( taken oue
of the equal parts) upon the Line of Superficies, and fo doing you
fhall find it fall far fhort of the number 64, the fecond term; there-
fore it is evident the work muft be done upon the Line of Superficies,
fo that Irake 32 fromthe Line of equal parts, and putting one foot
of that extent upon 64 in che Line of Superficies; I thereby open the
Seétor to fome one Line on the other leg of the Seétor, which iffueth.
fromthe Center. Then again, Itake the third number 24 out of the
| Line of Lines, andenter it Parallelly between the Superficies andthe
former Line on the other Leg ( in the manner that hath been fhewed.
before ) and I find it to ftay at 36 in the Line of Superficies: Sothae
Iconclude, as 32064, fo 64to 36in duplicated proportion. Thae
is, fo is the {quareof 64 namely 8) tothe {quare roos of 36 (namely

6)ina fimple proportion.

Z 6, Ha-

479 Ar. Sanel Fofterbis =~
| 6. Having two nambers, to find a mean proportional.

His is. performed by the joyne ufe of Superficies. and Lines: Let

the Numbers. given be 20 and 45, Firft, Icount che firftnumber -
20 upon the Line of Superficies, then J take the fame number 20 ont:
of the Line of Lines, wich this length I open the Sector from the
Point 20 in the.Line of Superficies,to fome one Line on the other Leg,
which iffueth from the Center 5 afterwards I count 45 the other gi-

ven number upon the fame Line of Superficies ; and from thence’

rake over the leaft diftance to the former Line on the other Leg, this”
meafured upon che Linéof Lines, gives 30 for amean proportional
between 20 4nd 45. |

7. Huving three Numbers given, whereof the two firft are fuppsfed to be
ina duplicated proportion, how to find a fourth, unto which the third
(hall bein the fimple proportion of the former ; that ts, As the [quare
root of the firft tothe (quare root of the fecond. — 3

His is likewife to be performed by the joynt help of the Lines of
TD sivet Bates and equal parts. Let che Numbers be, as 25 to 16, fo
40to what? The two firltrermsare to be counted upon the Line of
~ Superficies,becaufe between them the duplicated proportion ts contai-

ned ; and the other cwo muft be taken upon the Line of Lines, becaufe:

- between them is the {imple proportion contained. And to knowupon
what Line to work, I order the terms fo, as the first and third may be

of one kind, thus, as 25 t0 4.0, fo 16 to what? Now becaufe.25 upon o

the Line of Superficies( if two upon that Line betaken for 20 as we
do here cake it) is greater chan 40 upon the Lineof Lines, therefore
the lateral work mult be done upon the Line of Superficies. So that

Itake 40 oat of the Line of Lines, and.put over that length from 25. A |

in the Line of Superficies unto the Line of Lines upon che other Leg’

‘of the Sectors’ And the Seftor being fo opened, I count the third 4
Number 16 upon che Line of Superficies, and take over from thence
rothe fame Line of Lines. This length I meafure upon the Line of — _

‘Lines (from whence the fecond Term was taken ) anditreachethto 3 2.

So that as 25 (0 16, fuppofed co be in a duplicated proportion.one to
the other, fo is 40 to 32 1n the fimple proportion, whereof that other

is duplicated.

g, Having |

_ abseration of the Seitor. : oat
8. Having three Narbers, tofind a fourth ia aTriplicated proportion.

His work isto be done upon the two Scales of equal parts and So-
lids joyntly taken together. Let an example be, As§5 to 88, fo
135 to what4na triplicated proportion? Here the firft two terms are
of one kind, and fo the ewolatrer are alfo. Therefore (as before )
change the places of the fecond and third thus; As §¥ isto 125, fois
880 what ? The terms then being thus difpofed, you muft meafure §5
(taken outof che Line of equal parts ) upon 125 counted in the Line
_ ~of Solids, and you fheall finditof greater length chan 125, ‘whereby
-itisevident, that the work muft be done upon the Line of equa! parts. |
Accordingly therefore, take 125 out of the Line of Solids, and fet-
ting one foot of that extent upon the Number 55 counted*in the Line
of Lines, with che other turned about you mult open the Line of -
Sines upon the other Leg of the Seétor,asthe manner is. Which done,
fet one foot of your Compafs upon 88 in the Line of Lines, and from
thence sake the leaft Afdahes from the Lineof Sines: This diftance
being meafured upon the Line of Solids, fheweth 512, fo that, as 55
to 88 fois iz5 to 512 in atriplicated proportion. That is, As $5 is 0
88, fo is the Cubicroot of 125 ( namely 5 ) tothe Cubic root of 512
(namely 8) ina fimple proportion.

g. How to find two mean proportionals between two Numbers given.

i tee is done upon the Line of Solids and equal parts joyned in
nfe rogether ; lec che two extremes or given Numbers be 5 12,and
216, between which there are required two mean proportionals. Firft
from the Line of Lines I take 512, and with that extent I open the
Point 512 (accounted in the Line of Solids ) Itake over the leaft di-
ftance of the-Line of Sines, and meafuring the fame upon the Scale-of
| equal parts, Ifind it to make 384, This laft lengeh 3841 cake again,
and put it over from 512 in the Line‘of Solids to the Line of Sines. And
then [ take in length from 216 again ( counted inthe Line of Solids )
unto that fame Line of Sines; and meafuriog this length upon the Line
of equal parts,I find it to reach 288,which is the fecond mean propor-
tional, fothat I conclude, as 512 is to 334, fo 384 to 288, and fo 288
00 216.

Z a 10, Ha-

27x Mx, Samuel Fofter his

10. Having three Numbers given, whereof the two fixf? are fuppofed tobe
in a triplicate proportion, how to find a fourt. unto which thethird
fhall be in a fimple proportion, that w, asthe Citic root of the firft
tothe Cubic rest of the fecond.

: cy
i [eee is to be performed by. joynt ufe of Solids and equal parts.
Suppofe the three Numbers givento be 704, 297,98: and let ie
required to find.a fourth Number, unto which 98 fhall bear chat fim-
ple proportion whereof 704 to 297° isthe triplicated or Cubical pro-
portion.’ Firft, that I may know upon what Line the lateral work is to
be performed, I alter che order of the fecond and third terms, thus,
704,98, 297: andin this order I compare the firft and fecond terms
cogether ; ‘that is, I crake 704. our of the Line of Solids which ( in
this work ) is the proper Scale of it, and meafure it upoa the Line
ef Lines, which isthe proper Scale of the fecond Number 98; and _
thereby I find chat 98 is the longer, whereby it appears ( by the for-
mer dire&tions ) thatche Lateral work is to be done upon the Line of
Lines. Wherefore accordingly I take 704 out of the Line of Solids
and fet one foot of that extent upon 98, counted ( asthe fecond
term ) uponthe Line of Lines, and from it J open the Line of Sines
upon che other Leg of the Sector. And when the Sector is thus opened,
Itake the third term( 297 ) out of the Line of Solids again, and put
the fame over till it fit from the Line of Lines to the Line of Sines; fo
at fa(t I find one foot to ftay upon the foot 73 4 in the Line of Lines.
Whence I conclude, that as 704 is to 297, fuppofed to be ina tri«
plicated proportton one to the others Sois 98to 73 2, which ewo
Numbers do comprehend che fundamental and fimple preportion,
whereof that other is the triplicated.

CH AP.. Vv.
Of the Scale of Chords,

FT Hough the Scale-of Sines will perform all che ufes of the Lines of

A Chords, if every Sine be counted. by the double number ( as
Mr. Ganter hath fhewed ) yet becaufe miftakes are eafily made by thac

| numeration

| will be an Ark of 79.gr. and.

= avn 2 ‘ees ET Si af » dl e J = : yas he ~ 7. 4A

— gheration of the Sector. . 173

numeration; therefore ic will be more convenient to ufe the Chords
themfelves. Thenfesare chiefly, re |

1. To find any Chord, or to fet off any Ark or Angle, upon a Cire
cle, whofe Radius is given. : . hia

2. Having a Radius and any Chord or Ark affigned, tofind the Ark
which the affigned Chord fubtendech..

3, By having any Chord affigned, to find the Radius-according- ”
which the affigned Chord is to be eftimated,

1. The Line of Chords 13 numbred up to 90, and will therefore fet off or
meal ure any Ark within 9© gr. Bat if the Ark, be more, it mujft.da
it at twice or more times. :

> Et the Radius A D.be given, and lec. the Circle DABE be de-.

~_feribed with it; and let ic be required to fee off 79 gr. from the
Point AeFir(t therefore the Seétor muft be opened to the RadiusD A,
fetting one foot of that extent upon 6ein the Line of Chords, and
opening the other Leg till she Compaffes being turned about, do only
touch fome one Line on the.
other Leg of the Seétor, which.
iQueth fromthe Center: Then
from 79 upon the Line of
Chords, take the leaft diftanceto

the former Line on the other ‘se avanee sos cannes
Leg, che fame being fet upon the . ¥
Circle from A towardsB; AB D-

A.D B an Angle of 79 gr..
Suppofe again that upon the-

' {ame Circle I (hould fet off 139

gr. becaufe this exceeds 90; therefore I divide it into two teffer Ar-
ches; namely 79 and 6o: Firft therefore I fet off from Ato B 7935
and then from Bro E 60gr, more ; which together do make up 139 gr-
and fo shofe Arks that are greater, may be: fet off: at three or-four.

a. Having

Bei
-3

aaa
oe |

eRe ree 7

+) a ay, Satninel Potter ba

ee

towhat number of degrees that (hord anfwereth.

and then enter the Chord B A between the Scale of Chords and fome
one Line on the other Leg of the Seftor, which iffueth from the Cen-
rer ; and I find itto ftay upon 79°deprees : So that AB isa Chord of

79 deprees, being referred tothe RadivsD A. But if I had the large ~

Chord A Eto’the fame Radius D’'A, and would know to what Ark it
belongeth ; I mutt firft defcribe the Circle with the Radius D A, and

. then infcribe the Chord AE into the Circle; afterwards I divide the

Ark A E into any two parts, as at B, chentake the Chord A'B, and en-
ter 1¢ upon the Scale of Chords (the Seétor being fec to the Radius
AD ) and find it to fubtend 79 gr. Again, I take B Eand dothe fame
thing with that Chord, and find it to anfwer to 60 gr. Then laftly 1
add 79 to 6o, thefumis 139; which gives the whole Ark AE anfwe-
ting to the Chord A E,fo if A E had relation cothe grearer Ark AHE,

then you muft work as here you did by the Ark A BE, and when ehae 4

is found to be 139 take it out of a whole Gircle, or 360, fofhall you
find the greater Ark A H Eto be 221 gr,

3. Les AB le the Chord of 79 gr. given,
efimated to be fuch a Chord required.

then fhall 60 in the fame Line ( rightly taken over ) give the Radius
AD required.

But if the Chord A Ewere given, and-counted as fubtending the —
139 gr. then it will noc be foeafie, for if the Line of Chordsbeufed, —
there will be need of protraction, The better way therefore will be.
to do it upon the Line of Sines, thus, Take half the Chord A E,name- _
ly, AF, and count thavas the Sine of half 139 gr. thatis, 69% gr. fo —
putting over this length A F inthe Sine of 69 4 gr. you may from90 _

takeoff the Radius required.

» and the Radius to which it :

2. Having the Radius D A, and the (bord AB affigned, I would knwo <

m 4 my,
Therefore open the Line of Chords,asbefore, tothe RadiusA D,

| Bin the number of degrees is lefsthan 90, therefore the work
will beeafie. For if A Bbe put over in79 inthe Line of Chords, —

Note that if itbe required to open any two Lines of the Segtor ‘to i
any number of gr. lefsthan 90, or if whenthe Lines be opened, it be —

reqnired to know at what Angle they and. Then it will be the bet

way

| fy Ue a
ws

_ Mr. Gunters Book, concerning Projections of
ling by the Se€tor,and other ways by a‘donbleiTangent of 45.) Hereun- -

| cording to the leaft diftance,
| is equalco the

a

altevadianiof the. Saébers, 75
way to ufethe Scale of Sines in this manner. Becaufe every Line hath
a Pointat the very extremity of ir, therefore if you.take-any Sine out

~ of the Scale of Sines, without the forementioned doubling of the

Numbers + and from the extremity of one Line, do enter chat Sine
according to the leaft diftance from the other Lines. then thall:thofe
two Lines ftand at the Angle required. |

Or if from the extremity of one-Line, you take the leaft difance
of the other, the fame meafured upon the Line of Sines, fhews the
Angle at which thefe two Lines ftand. Orfor moftof the Lines, if ie
be needful, you may ufe the way that Mr. Gunter fhews, Lib. 2. c. 2.

Prope]. Arts.

CHAP. VI.

Of the Tangents and Sécants.

Efdes other ules of thefe two Scales, they ferve for Projectionsy .

and for Dialling to any bignefs greater or fmaller, (of which fee
the Sphere, and of Dial--

co thele two Propofitions rend.

+. Having any Line given, aaakgawn Tangent, or Secart : Ts find the
Radius belonging to it.

~\ Uppofe I would know to what Radius any given Line fhould repre- -
fent a Tangent of 40 gr. Itake the Line given, and fet-one foot of

it in 40 inthe Scale of Tangents : And from: it I open, che Sector to :
{ome one Line on the other. Leg which ifueth from the Center, ac-
Then from the Tangent of 45 (which

Radius) Itake the leatt diftance to the former Line on.
the other’ Leg, and that length is the Radius required. |

The fame work isto be done by the Secants, where the Radius ts une -

| “de rftood to be at the beginning of the Scale...

22 Having 4

Bee

176 MrSamnud Bolter his

; 2. Having the Radiws, to find any z. angent or Secant belonging thereto. Kae :

His is eafie and like the former, if ‘firft the Seétor be opened to
the Radius, cc.

COROLLARY.

By thefe two Propofitions a way wsay be found, how by having a Tangent, to
find any Secant, or Sine.belonging to the fame Radius.

Or firft, you muft find the Radius, and then the Secant, yor Sine af-
§ cerwards. So alfo by having the Secant, may be found the Sine or
Tangent, or having a Sine, a Tangent or Secant may be found,

And foto a Radius of any length, you may proportion any Sine, ©
Tangent, or Secant. And note, that for pricking down the hour Points
in Dials, the Tangents of 15, 39, 45, and 60, willbe of frequent ufe.
And if the Tangent or Secant Scales be not far enough extended,
Mr. Ganter hath given rules how to enlarge them. :

CHAP. VII.

How to [apply the Aderidian Line i Line of Rambs, by the Seale

of Secants.

Ti Aow to make a Sea-Chart, after Mercators Projeftion.

1 Fase Propofision is the chief, that the Meridian Line upon the —
Sector doth, concerning Navigation, and therefore Mr. Gunter
mikes it his firft Propofition. And thisis performed by the Line of Se-
cants. For if it were required to project fuch a Chart as is in his Book. —
Having drawn his Line A B,and crofled ic with the Parallels 50 and 50,
atright Angles,you muft chen take the Secant.of 51 gr.from the Sector
opened to the length you defire, according to the leaft diftance, (the
manner whereof hath been fhewed enough afready, ) and fet from 50 |
to §1, on both fides of the Chart, and draw : and 51: Again,take |
the Secantof 52, fromthe Seftor, and fet it uponthe Chart from 51

to

Se ek

mea ao, = aie, 2 ll ok 0 eee ie)! ee
(5 eR = gh
Ehf « ’ te .
ao é
” e 4
{

| alteration of the Sector. fa
to §2, andfo drawthe Parallel 52, 52, And thusyou may draw all
thereft of the Parallels. Then for the divifions or Meridians of che
Line B C, they are all equal to the Radius, if cherefore you take the
Radius, and turn it above and below, you fhall make che {paces or di- —
_ftances of che Meridians, fuch as in che bottom of his Chart are figured.
with I, 2, 35 4, 5; 6. |
_ Thefe degrees thus fer oa, may be fubdivided into equa! pares,
which inthe graduations.above and below ought fo tobe, but inche
graduation upon the fides of the Chart,they ought as chey grow higher,
ftillto grow greater. Yet the difference is fo {mall, chat ie cannoe
produce any fenfible errour ehough the divifions be ali equal. Divide
them therefore equally into 60 minutes, or Englith miles, or into 2a
Leagues, or into 100 parts of degrees, as thall beit be liked of,
If aliecle more Curiofity fhould be ftood upon for the graduations
of the Meridian, then initead of the Secants, 51, 52, 53. you
may take 0%, §1 4, 52 2; always halfa gr. lels chan che Catietude chac
- fe tobe pucin. nee
2. The ufes of the Sea-Chart, and forme other Propofttions that concers
Navigation, are {et down by Mr. Gunter lib.2. cap. 6. of bss
Seétor, which may be bere alfo done. ne

He manner of working upon the Chart ( which is the beft way )

his dire&tions will fhew, and how to work them upon this Seétor,

the former directions in this Treatife will be fufficient. So by this

means the ufe of the Meridian Line is fully fupplied, becaufe each De-

_ gree may be very large, which in the other Seétor could not be fo

- withous each pare many times repeated, which ching will produce as
much erroar as thisway by theSecantscando. __

CHAP. VIII.
The ufes of the Line of Verfed Sines,’

He Ufe of it is generally as much asthe fourch Axiom of Spberi-
cal Trigonometrie. Wherefore,I will irft fhew how the twocafes of
that Axiom may be performed by this Line. And afterwards how fome

particular Problems of more frequent ufe may be performed.
7 Aa : 1, Having

998 | Myr. Samuel Fofter bis

1, Having two fides of 4 Spherical, Triangle, and the Angle compres
bended, to find the Bafe. | ,

Mrit; Gnd the fum and difference of the two fides or Legs, then
count that {um and difference upon the verfed Scale,and with your
Compaffes, cakecheir diftance, with this diftance from the end of the a
verfed Line, open the Sedtor to fome one Line on che other Leg,which

- #{fuerh from the Center afterwards upon the verfed Line, count the

Angle given, and fetting one foot in that number, take the leaft diftance
from it ro the former Line on the other Leg, this length being added
éo the difference of the Legs upon the veried Scale, gives the Bafe
required. ;

“So if che Legs were 38 gr. 30 m. and o§ gr. their fum would be 13 3

yr, jom, their difference would be 56 gr. 30 m. And now the diftance
of thefe beingtaken, arid che Se@or opened as is prefcribed, fuppofe —
the Vertical Angle were 75 gr. from75, therefore take the lealtdi-
ance, co the former Line on the other Leg, and fee that diftance on
the verfed Scale from the former difference 56 gr. 30m. numbred
thereon, it will fail upon 84 gr. 42 m. which isthe quantity of the Bafe
of che Triangle. |

} An Example. 2
T wo places differing both in Longitude and Latitude, to find their Diftance.

Ec the cwo places be London and Hierafalem, the Latinde of Lone
don is 51 gr. 30m, the Latitude of Hierufalem is 32 gr. che difte-
rence of Longitude 47 gr. their diftance is required,

The fum of the Complement of the two Latitudes is 96 gr.— om.
‘Their difference nV 19 ——30,

Take the diftance berween.96.gr. 30m. and 19 gn 30m, with this

extent of che Compafles, opea the Seétor from the end of theverfed —

Scale; the Se@tor thus refting, cake che -neareft diftance from
che difference of Longitude 47 gr. this diftance being applied to

* 79 gr. 30 @, On the verfed Scale, wiil reach to 39 gr. 14m. the diftance

required.

2, Having

he ie SY cual nA Tr, a i Pale ad
f ve) as bia, ape Be ; /
re eee Shenk

alteration of the Sector. | | | 179

ae Having the three fides of any Spherical Triangle to find the Vertical

Angle. .
Hat Angle that is required, is called the Vertical Angles The fide
: prpcie’ to it.is called che Bafe; and the other two fides are called
the Legs. ©
1, Find the fum and difference of the two Lege, chen count both che
fum and difference uponthe Scale of verfed Sines,and with this diftance
taken in your Compaffes, from the furcheft end of che verfed-Scale,
open the Sector to fome Line onthe other Leg, which iffueth from the
Center, asthe manner hath been. Afterward take the diftance from
theforenamed difference of the Legs to the Bafe, counted upon the
_ fame Scale, this diftance ts to be applied to the two Scales before open-
ed, and now appointed for the work, fo as that the Compafs foot be
removed upon the verfed Line, till the other being turned about may
jufttouch the former Line omthe other Leg, and where the foot of the
Compafs ( with this condition ) fhall ftay upon the verfed Line, there
fhall you fee the quantity of the Vertical Angle required. Or if after
the Seétor be opened, you take the diftance, not from the difference,
but from the fum of the Legs co the Bafe, that diftance will find ehe
Supplement of the Vettical Angle, whichin fome cafes. is mot re-
uired. |
: Soif thetwo Legs were 38 2 gr. and og m.che Bale 84 gr. 4.2 m, the
Vertical Angle willbe found tobe7§ gr» Or the Supplemene will
be found to be 105 Degrees. : 7
Thefe two Propofitions thus generally propounded, do. ( in brief )

fet forth two of the principal ufes of chis Scale of Verfed Sines. And -

cochele two I will add one more, which is done without opening the
Seétor atall.

3. Having a proportion to be wrought in Sines alone, whereinthe Radivas
leads in the proportion, bow to find a fourth proportional Sine upon
this Verfed Scale. |

Akethe fum and difference of the fecond andthird Arks; coune —~
chem upon the Scaleand take the difference of chem; if youfee |

this diftance equally remote fromoo.upon:the Scale on both fides of ir,

you fhall fee both che feet of the Compaffés to ftay upon the fourth
Aa 2 proportional

: 4
t+ r é < fa.

186 Mr. Samuel Fofter bis
Proportional Sine required. Suppofe the proportion to ftand thus. As

the Radius, to the Sine of 60 gr. fo the Sine of 40 gr. to what Sine?
The fum and difference of the two given Arks, 60 and 40 are.100 and
20. Ttake thediftance of thefe twocounted upon the Scale, and fer
it equally diftant cn both fides from 90, and i find it to tay in 33 2
from9o, Wherefore I conclude that as the Radius is in proporeion to
the Sine of 6ogr. fo the Sine of gogr. to the Sine of 33 ger.

els for the former general Problems, that their ufefulac{s may be more |
_ mauifeft, I will here add three Propofitions deduced from thems,
which are of daily ufe, and by ether general Inftruments performed with
much difficulty.
- The firft thall be, To get the Suns Azimuth.
The fecond, 70 find the hour of the day. ;
The third, To fiad the Suns Altitude at any bour.
The firft of thefe is,
a Having the Latitude of the place, the degree of the Sun inthe Zo-

diac,and the Sens Altitude above the Horizon, to find the Sans Azxi=
month either from the North, or frome the South.

Ban. this Propofition is fo very ufefulin many particulars, there-

fore principally is the Zodiac Line annexed to this verfed Scale,
and therefore alfodo!I fer it inthe firft place.
' This fallsunder the fecond general cafe delivered before 3; Thetwo
Legs of the Triangle are the Complement of the Latitude, and Com-
plement of the Suns Altitude. The Bafe isthe Suns place in che Zo-
diac, taken fronithe beginning of the Line. The Vertical Angle in
thefe Northern Latitudes, is the Azimuch from the North.

Take the Sum and difference of the Complement of che Laticude,
and the Complement of the Suns Altitude, and count this fam and dif.
ferendeupon the verfed Scale, and with your Compaffes take their di-
tance, wich. that diftance open the Se&tor to’ fome one Line on the
other Leg, which iffuech from the Center, from the end of the ver-
fed Line. Afterwards take the diftance from the aforenamed difference,

tO

OEP E ee Pal | pCR RIA T TVR. SMO MeE TRHIN pe Rei oye Te LAS 9 PUPA Dail Oo a ae a

alteration of the Setter. — r8r
to the Suns place, and encer it berween the former Line onthe other

_ Leg,and the verfed Scale, and note the Point of the verfed Scale on

_ which the foot of the Compafs ftays, the fame Point fhews the Azi-
_- muth from the North. Or when the Seétor is openec, take the di-

ftance fromthe fum, coche Suns piace,the fame entred as beiore, will

give che Azimuth from.theSouth. Nore chac if the fum do exceed

18o0gr. then you are to account 170 as 190, 160 as 2C0, 150 as

210, Ce. :

Example.
ip the Laticude 5 gr. 30 ms. TheSunbeing inthe begining of Ts-
rus, andthe Altitude 35 gr. The Complement of the Latititude is

383 gr. The Complement of the Altitude is 55 grs The fum of thefe
two is 93 gr, 30 m. their difference is 16 gr. 30 m. I takethe diftance:
of thefetwo counted upon the verfed Scale, and with icdo open the
Sector to fome one Line on the other Leg which iffueth from the Cens:
ter, fromthe end of the verfed Scale: then 1 cake the diftance from
the fum, 93 4 gr. co the Suns place 00 Taurus, and enter it uponthe
verfed Scale, cothe former Line on-the other Leg, and find the foor
of the Compafs to flay at 60.gr. 42m. which is the Azimuthre.—
quired, :

COROLLARY

The fame things given, To find the Amplitude of the rifing and fetting
ef the San.

_ WE yon fuppofe the Sunco have no Altitude above the Horizon, and
Sfothe Complements of it to be 90, and then work by the former:
_ precept, the Ar(ftway fhews the Amplitude fromthe North, and che>
fecond way fhews the Amplitude fremthe South. And if either of
thefe two numbers be numbred from the middle of the Line noted:
with 90, you fhall have the Amplitude from the Eaft or Wet. Soro:
_ the beginning of Zs#rm,1 fhall find the Amplitude to be 108 gr. 41 m.:
fromthe South: 75 gr. 19m. fromthe North: and 18 gr. 44 m.from:

the Eaft or Welt.

A fe-.

82 “Mr Samael Fokter his

A fecond Example is, ua

The Latitudes of two places, and their diftance being given, tofind their
Difference of Longitude.

YF Ecthetwoplaces be London and Hiernfalem, the Latitude of Lone —
| it st gr. 30 m,0f Hiernfalem 52 gr. their diftance 39 gr. 14m.
and their difference of Longitude required.

The fum of the Complements of the two Latitudes, 96 gr.—30 m,
Their difference 19-—-—30

Take the diftance between this fum and difference, and open the
‘Sector fromthe end of the Verfed Scale, then take the diftance from
the difference 19 g7.30 m. to the diftance given, viz,-39 gr. 14m,
where thac fits over from the Veifed Scale, which willbe at47 gr. is
the difference of Longitude required. The next thing is,

Having the Latitade, the Sans placein the Zodiac, and the Altitude
abu: the Horizon, te find the hcur from the South.

eh alfo falls under the fecond general cafe before delivered.
The two Legs of the Triangle, are the Complement of the La-
titude, and the Suns diftance from the elevated Pole: the Bafe is the i
Complement of the Suns Altitude above the Horizon; the Vertical —
Angle in thefe Northern Latitudes is the hour from the South, or
Mid-day. | : i
Pirf, Count the number of gr. from the beginning of the Scaleto Uy
the Suns place, this number compare with the Complement of the ie
_ Latitude, and find che fumand difference of them. Then upon the
Verfed Scale count this fum and difference, and take the diftance of 5
them, and with that diftance open the Seétor, as is prefcribed in the’
former Propofition. Then take the diftance of the Complement of
‘the Suns Alticude, and the afore-named difference; and enering it”
upon the Verfed Scale from fome one Line on the other Leg, which —
iffueth fromthe Center, you fhall find it to tay upon.a number of
degrees, which turned into time, gives the hour required. One hour —
aniwersto 1§ gr. one degree makes four minutes of time. Notehere —

alfe,

ae

ae |

|:

~~ nal Ark from th

‘alfo, that if the fam do srife to above 180 gr. youare then to accoun®
170 a5 190, 160 as 200, 150 a5 210, aswas before intimated.

Example, From the beginning of the Scale to 00 © is 784 gr, this

I compare with 383 gr. end J find the fumof them tobe 117 gr. and
the difference to be 40 gr. then 1 count thefe two numbers upon che
~ Verfed Scale, and take cheir diftance; with this diftance, I openthe
“Se@or from the erd of the Verfed Scale, tofome one Line onthe
other Leg. Again, I take the length from 40 gr. (the fore-named
difference ) to $5 gr. the Complement ot the Altitude ( which I fup-

pofe to be the fame that was in the former example ) and this lengrh 1.
he Verfed Scale, from the former Line on the other Leg, .

enter upon t
and find the foot of the Compafles to ftay in 46 gr. 48m. This Ark

contains 3 hours and 7 3 ™. of an hour, and fo mucii isthe hour at thae

time from noon. Ifthe Altitude therefore were obferved in the morn-

ing, it mult be 53 . palt 8 of the clock ; if in the after-noon,it is 7 a.
paft 3 a clock. mae uh
Or if [hed tcken the length from the fore-named fum 117 gr. to
$5 gr. the Complement of the Altitude, and had entered the length,
as before, I fhould then have found che Supplement of che former,

name!y,133 gr. 12m.
hours and 53 7. whic
were taken in che morning,
12, namely, 3 hours 7 minutes,
cion were made in the afternoon.

1, Corollary. .
To find the Semidiurnal and $ eminciturnal rks. »

“FE you fuppofe the Altitude to be ©0, and fo the Complement of it to;
9 be 90, and then work by chis Precept, you fhall find the Semidiur-

e beginning of the Line, and the SeminoGurnal Ark

umbered in degrees, and each of thofe

from the end of the Line, n
and doubled, will thew che length of the.

turned into hours and minutes,
day and night. } i
om the degrees of the Semidiurnal Ark, you take 90, you

And if fr
fhall have the Afcenfional difference in deprees; or if you take fix
hours outof the Semidjurnalhours, you fhall have the fame Afcenti- -

onal difference in time. .
Examples.

- alteration of the Seétor, A aes

which is the hour from mid-night, namely,eighe-
h is the hour of the day, if che Suns Altitude-
or elfe che Complement of that hour to.
isthe hour of the day, if che obfervae—

ay street
mots ASS CREATOR! Le Ph et OPN ER
ie

184 Me Samuel Fofter ba |

Example. In the fame Laticude, and the oo gr. of U, the Semidi- —
urnal Ark willbe 104 gr. 49 m. Thefe doubled, make the length of the
day 209 gr.38 m. Or the fame turned into hours, make 6 hours 59 |
Minutes, and chefe doubled, make the length of the day 13 hours
58 minutes. The Seminoéturnal Ark is 75 gr. 11 . or4.hoursos
minutes. Thefe doubled, make the length of che night 150 gr. 22 m.
or 10 hours 1 + minutes of an hour. ae |

‘ The Afcenfional difference is 14.¢r. 59 m. or oo hours 59% of a
our. - |

oe) eee Fe ee eR Ie EEN ec eo mR ty te he ORL. RAPON IN tN oe MOORE TROBE D7 ee Sgt PRS ,
: emi ache fics " Pass isos pias - OT NS ed

2, COROLLARY.
To find the moment of tine, when the Crepufculum begins and ends.

F you fuppofetheSun to be 18 gr. below the Horizon, and fotake
Liem the former difference of the Legs,downto 108, and enterthae
length as before, you fhall find wha time fromthe mid-day thetwo —
twy-lighis begin and end.
Example. In the beginning of Tawres, the morning twy-lighe.be-
gins 139 gr, 40 m.beforenoon, chat is 9 hours 182 Minutes, or at
-41 3 Miaures paft two a clock inthe morning, and che evening ewy-
ie light ends 139 gr. 40 ms. after-noon, or at 18 ; paft nine a clock at
~ night.

3e5 COROLLARY.

T he Suns place being affigned in any Polat of the Zodiac, to fad bis Ale
titude at all hears. “aN es “We

| Problem falls under the firft general Cafe before delivered. _
The two Legs of the Triangle are the Complement of the Lati-
tude, and che Suns diftance from the Elevated Pole. The Angle in-
tercepred between them is each hour from the South, whofe Altieudes
ae required, The Bafe is the Complemenc of the Altitude foughe
or. ; ie
Firft, Find the fum and difference of the Complement of the Lati- _
tude, and che Suns diftance from the elevated Pole, count bothchis
fury and difference upon the Verfed Scale, and cake the diftance of
them, and open the Sector to fome one Line on the other Leg, which

iffueth

i

ink ae TO eri

5 fee 8 Oa Se et Oe ama OR EE ET Aes riper a eee a
Epes. i aia Wh hs 6 a Sia te set AL CIE Ge 2s
teks as oe ila’ tie he , ' a Pe ce: fe
7 Maes ¢ tel 2 ‘ 4
4 r fs - : \

alteration of the Seto. . —-185

| iffueth-from the Center: ro thaed iftance, from the end of the verfed
_ Scale. Then count every hour uponthe verfed Scale (allowing 15 gr.
_ toan hour ) and from thofe hour- points take the leat diftance to che

former Line onthe other Leg,thefe diftances being fee from che afore-
named difference of the Legs outword upon the verfed Scale, will
give che Complement of the Altitudes to each feveral hour from the
Meridian. Or if they be numbered from go in the Scale, tothe foor
of che Compafles neareft to 1g0 upon the Scale, you fhall have the
Alcicudes tchemfelves.

Note, chat if you go quite through every fifteenth degree, or every
of the cwelve hours upon the Scale, you fhall go beyond 90,,and thofe
degrees beyond go are the profundities of the night-hours, che Sun
being in chat degree of the Zodiac. And they are alfo Altitudes of
the hours for the Suns being in the oppofite degree of the Zodiac, So
that one opening of the Sedtor will ferve co find che Altitudes of all
the hours in any two oppofire Signs or Points in the Zodiac, Nore
alfo, thot the difference of the Legs isthe Complement of the Suns
Alcicude at 12 aclock at noon, and che fum of them being diminithed

by 90 gr. is the depth at mid-night or the mid-day Altitude of the
_ Sun, when he is in the oppofire Sign or degree.

Note laftly, ( as formerly ) that if the fum of the two Legs do
amount to above 180, you mutt then count 170 fer 190, 160 for 200,

150 for 210, &c. as was noted before.

‘Becaufe this Propoftion » fo frequent in ufe for the making of Tables of

the Suns Altitude in every Sign, or any Parallel of Declinationywhich -
ferve for drawing particular inftrumental Dials, as Quadrants,
Rings,-and Cylinders, and for all other parpofles alfo, 1 wih therefore
- add one example at large, t0 mare it the more plain. |

Example. Inour Latitude 51 gr. 30 m. the Sunbeing 00 Taurus, I

would know the Suns Altitudes at every hour of the day, and the pro-
_ fundities of che Sunat every hour inthenight. The Complement of
the Latitude is 38 gr. 30m. and the Suns diftance from the North
Pole is 78 gr.30. the fum of thefe is 117 gr. the difference of them

is40 pr. Firft chen, I count thefe two aumbers upon the verfed Scale,
and take their diftance, with this. diftance Lopen the Seétor to fome
one Line onthe other Leg, which ifluech fromthe Center, from the

end ef the Verfed Scale, Then I count 15 upon the verfed Scale,
Bb and

a TALS SOAR TE Bee Pe

Oey Une Nie Meee een ey
SFT ae ed — eye ey ae

186 — Mr, Samuel Fofter his

and from thence I take the leaft diftance, to the former Lineon the ~

other Leg. One foot of this diftance I fet upon the difference of the
Legs ( which was 40 degrees.) The other I fee forward uponthe ver-
fed Scale, and where it falls, it (hews 41 gr. 48 m. the Complement of
the Suns Altirude at 1t and 1aclock, or counting it from 99, it fhews
48 gr. 12 m. the Alcitude ic felfac 11 and 1a clock.

So again, I count 30 upon the verfed Scale, and take the leaftdi-

france to the former Line on the other Leg, and fet one foot upon the
difference of the Legs ( viz. 40, gr. ) the other forwards upon the
verfed Scale. [find itto fall upon 46 gr. 48m, which is the Comple-
mentof the Suns Altirude at 10 and 2a clocks or counting it from 9e,

it falls upon 43 gr. 12 m the Alcicude itfelf, av10 and 2aclock, or

from go gr. it falls upon 43 gr. 12 m. the Alcicude tt felf.

In the fame manner taking the leaft diftance from 45 gr. to the for=
mer Line on the other Leg, and fetting one foot of that diftance tothe
difference of che Legs,you thall find the other to fall upon 54 degrees,
which is che Complement for the Altitude for 9 and 3 aclock.

And fo working from 60,the Compaffes will they che Complement.
of the Altitude 627. 29 m, andthe Alcicude ic felf 27 gr. 31m. for
the hour of 8 and 4. ess a

Andat 75 decrees, having with your Compaffes taken the leaft di-
ftance,and fet itas before tothe difference of the Legs, will give 18 gro ty
18 ms. for the Alticude of 7 and 5 aclock. _

And at 90, or 6 aClock, the Altitude will be 9 gr. 00 m. 5

So working {till inthe fame manner, from 105, upon the Verfed
Scale, you fhall tind your Compaffes co reach beyond 92, namely, to
90. gr.06 m. for 5 inthe morning, and 7 after noon. From which, if
youtake gogr. the remainder fhews how much the Sun is below che y:
Horizonat 5 in the morning, and 7at night; namely, 6 minutes, Or
it fhews how high che Sun will be, when it is in che beginning of Scores
pio, the oppolite fign to Tamrm, at7 in the morning, and at 5 after s
noon: and doing the like from 120, you fhall find the Compaffes to
fhew 98 gr. 23%. from whence caking go, there will remain Sgr.
33 m. for the Suns profundity at 4. in the morning, and 8 atnight, the
San being inoo of Tawrws, or 8 gr, 33 m: for the Suns Altitude at 8
inthe morning, and 4 after noon in 00 of Scorpio, Ati35 gr. the —
profundity for 3 and 9, or the Altitude for 9 and 3, willbe 15 gr.58.m
“At Tyo, che Protundity for 2 {and 10, or che Altitude for toand2.

willbe zrgr, gt, At1S5, ce Profundity at Tandt1 inoo of Taue
ie Ty

‘

sh Lierade of the Sector, " | 187

rm, or the Altitude of 11 and inooof Scorpio, willbe 25 gr. 40 m.

And laftly, whereas the difference of the Legs was found 40 gr, by
what was formerly intimated, the fame 40 degrees, do fhew che Com-
plement-of. the Suns Altitude at 12 aclock, whenthe Sunis inoo of
Sor pi.

By this appears the manner of refolving this Propoficion, and how

thefe Tables may be made to other Signs or Points of the Ecliptick,

or Declination from the Equinoétial. nS ge .

Note alfo thar the work may beginwith the Winter figns, and end
with the Summer, as here it may begin with Scorpio, and end with 7 aa-
us, thus. From the beginning of the Line tothe beginning of Scorpio,

py are 101 gr. 30m. This diftance compared with 38 gr. 30m. makes

the {um 140 gr. and the difference 63, the Complement of chis diffe-
rence is 27, the Altitude of 12 at noon in the beginning of Scorpio,
andthe Excefs of 140 above 90 is 50, which gives the midnights Pro-
fundity at the beginning of Scorpis, or the mid-days Altitude in che

_ beginning of Tanrus. And if you work for the other hours (as in the

Jaft example was largely fhewed ) you fhall find che Altitudes pointed
out by the other foot of che Compaffes, for each hour in 00 of Scorpio,
untill you come towards 90,and when you come beyond 90, the Excefs —
fhews the Profundity for the reft of the hours of thenight in Scorpis;
but the Altitudes for the anfwerable hours in the beginning of Zaa-
rus. And fo allother Signs and Parallels of Declination.

Thefe arethe particalars in which I intended to exemplifte, becaufe

cheir ufes are more frequens than the others are.

By the like work, havingthe Deelimagion and Reclination of any

Plane, may be found. Firft, The Poles Altitude above the Plane; then

in proportions in Signsalone may be found, the Planes difference of Lon-
gitude, with the departure of the Subjtyle from the Vertical Line: andby
thefeche Dial may be made, and the Lines placed in a right pofitione
So,by che like work, having the difference of Longitude of any Ci-
ty,or remarkable place from yours,and the Latitude of the fame place,
you may find in what Pofitiona Plane is to be fet in your. Horizon, in
refpect of Declination and Reclination, or Inclination, that may repre-
fent the Horizon ot the fame place, and accordingly you may put on

~ the hours that belong to that Plane or place, with all the other furni-

cure whereby the Policions of the San, in refpeét of the place, may -

be reprefented to your view upon the Plane.

- Andbefides thefe, there are many other particulars which may be
Bb 2 ; ‘performed

188 | Mr. Samuel Fofter bis

performed nponthisScale, namely, all chat fall under the fore-men-— a

tioned tourth Axiom of Spherical Trigonomecry.

a

~~

“ C H A P, , X. -.
Of the ether Scales onthe edges and fpare places of the Sectors

Henthen the Seétor is opened intoa ftraight Ruler, chen will

VV che divilions of the outer edges of the two Legs, and upon the
two flat fides, ( which in the former ufage of the Sector appeared to
be divided or broken) be made up, and become as entire Scales.

1, Thofe on the outer edge are che three ufual Logarishmetical
Lines of Numbers, Sines, and Tangents. The ufe of chofe are fhewed
moft largely by Mr. Ganter in the ufe of che Crofs-ftaff, and therefore
I fhall need to fay no more of-them. |

2. The two Scales of Foot and Inch meafure, upon one of the flat

fides, will (erve co meafure fmall lengehs, chat reach not above one
foor, or ele co make longer Scales of feotor inch meafure, whereby
greater lengths may be mealured,

And this is all I thatl needto mention of thefe two Scales, the ochers
that remain, require fomewhat more to be faid of them.

iA

CHAP. XI.
Of the two Scales of wine and Ale meafure.

It is here fup- § Wine Gallon contains 231 2 Cubical
pofed shata 2 Ale Gallon contains 288 § Inches.

Hefe two Scales ferve for the {peedy gauging of Wine, Ale, or -

Beer Veffels ; and therefore you mutt firtt prepare a Staff of con-
yenient length, whereby co take she Dimenfions of any veflel. And
upon che Sraff. | :

Fcr Wine, you muft fet on thelength of 4 foot and 43, parts of an wh

inch, which length is to be divided into 36 equal parts, with decimal

fubdivifions futable coeach part, and continue the fame parts quite
through che lengeh of che Staff. For

a

el uber. 8 os) Rat RR al

Ee ees Ed a et ee ENE TD, OE aS Se oh eee Me RTs m8
Dt eMart as tet LOT DTT oro re Me Nee RST pe) Sey ee a tea so
Z tz ~ i :
c 3 ‘

| alteration of the Sectors pag
For Ale, fet onthe lengeh of 4 foot and two inches juftly, and di-
vide che fame into 25 equal parts, andfub-divide thofe parts decimally
into as many fub-divilions as thofe parts will contain.

Wich chefe Scales you are co meafare the Wine or Ale veffels, that
is, you muft eaketheir Diameters at che bung and head, and meafure the.
lengeh of them-each with his proper Scale.

~

Then to find the Content inGalons.

Count the length of the Diamecer at the bung, uponthe proper
Scale ( that is, upon che Scale of Wine-meafure apon the Seétor, if ie
be for Wine, or upon she Scale of Ale-meafure, if it befor Ale ) and.

. taking the famein your Compaffesy apply it to che Line of Superficies,

fetting one foot inche Center of the Se&tor, and mark where the other:
foot falls, and noting the number, write itdown twice. In this work .
the whole Line of Superficies is fuppofed to contain but ten parts only.
Again, countthe Diameter atthe head, uponthe proper Scale of the -
Sector, andapply ehat length likewife tothe Line of Superficies, and .
nore what number it fallsupon, end write it under the two former, on-

- Ty once:- then: add rhefe three Numbers together, and keep the fum, .

Afterwads going tothe Line of Numbers. Say,
As 2, to che length of the Veffel. | ‘
So.the former fum, co the content of the Veflelin Gallons.
Example.

Suppofe a Veffel whofe Diameter at the Bung, contained 22**

' ‘partsof the Scale of Wine meafure: the Diameter at the head 187*

of the fame parts; the fength of the Veffel 30 partsof fuch a Scale
as is formerly prefcribed for Wine meafure, from which :thefe mea-

fures here fuppofed aretaken. —
The firft number of the Bungs meafure I take from the Scale of -

~ Wine meafure upon the Seftor, and applying itcothe Line of Super-

ficies, I find it shen to fall upon 42*, which number I fetdown .

twice.

Then again I take the head number from the fameScale of 4.°70 »

| Wine meafure upon the Sector, and when it is applied to 4,70
_ the Line of Superficies it reacheth co (about) 344 which I fer 3. 25 |

under the two fermer numbers in right order,asinthe Margin 53. ‘s ‘
| the

yess “adr, Samuel Folter his

the fum of thefe three Numbers is 1245. Then upon the Line of —

Numbers I work this proportion; — | a)

As 2istothe fum 12 **: fo isthe length 30 to 190 ‘near upon. So
thae the conrent.of fucha Veflel is near 190 Wine Gallons. And if it
had been computed by Numbers, the concent would be about ;*; of a
Gallon lefs. fiegte

q If the fame Veffel were meafured by the fore-mentioned Staff
made for Ale meafure, the Diameter at the Bung would be 18 of
thofe parts, the Diamerer at the head 15, the length of the Vefiel 24.
And the two former Diameters being taken upon the Scale of Ale mea-
fure upon the Seéter, and applied tothe Line of Superficies will pro-
duce the fame three Numbersas before. Then as 2 tothe fum of them

= ee

1245: fois 24the length to about 152, which isthe contentin Ale.

Gallons. -

CHAP. * XII.
Hew to perforns the fame work of Gauging by Feet or Inches.

Or this purpofe there are two Scales upon the inner edge of one
of che Leggs of the Seétor, called Feet and Inches, which Scales

cannot be true Feet or Inches, as by their length will eafily appear, a
but for this work of Gauging ( whereunto they are chiefly intended )
they are co reprefeat Feet and Inches, and accordingly are here called a

reprefentative Feet, and reprefentative Inches. |

Now totakethe Dimenfions of a Veflel, namely, the length with

the Diameters at head and Buhg, you mufthave a ftaff divided,
Either into true Feet, and each foot decimally fubdivided :
Gr into true Inches, and each Inch decimally again fabdivided-

Thefe: Scales may be made from the true Foot and Inch Scales ins Ae
fcribed upon the flat of the Seétor,as was intimated before inthe tenth

Chapter. . : Ge
Eicher of thefe two Scales will perform what is here intended.

Wich one of thefe Staves ( which you have moft mind to) you are —
to meafure the V effel ;and Knowing of what Numbers each Dimention y

is, you may caft upthe content of the Veffelthus.

To ‘e

three Numbersand keep the fum.

- Nomber of the Veffels length inFeettoa 4sb. Andas I to

‘alteration of the Setter. jl ‘391
sea oF find the content in Gallons.

Count the Number of che Diameter at the Bung, upon the Scale
of reprefentative Sh iat - Lupon the Sedor, and take thet length and
fet it upon the Line of Superficies from the Center forwagds, and fee
what Number it there falls upon, write chat Number-twice. | Remem-
ber here again that che Superficial Line in this work is to be efteemed
of 10 parts only. | So do‘alfo with the Diameter at the head. and
write that Number under-the other two once only. And add thefe

. Feet
Then look what the Veffels length wasin Ne hae count the
{ame number upon the Scale of reprefentatives hes $and whenyou

have taken it off from thence, meafure it,

Either upon the Line of 3 hee Smeafure, and fo what Number ie

| reacheth thereon. Then upon the Line of Numbers fay, as 2 isto this

laft Number, fo is she fum before found, to the Number ot 5 es é.

Gallons contained in the Veffel.
Or without mesfuring it ufon thofe Lines of Wine or A'e Mea-

fure, youmay do inthis matiner. Find the fum of the three Numbers, |
asbefore. Then uponthe Line of Numbers fay, as 2 tothat fum, fothe

7, 481 for Wine, : - ¢ Wine >)
dF colfotoAte; ¢fo the 4th Number to the content in $ re
Gallons. ;
If the Veffels length be taken in Inches, then thus: As 2to the
{um, fo che Veflels length in Inches toa 4th. And again, as 10 to
$ 6. 234 for Wine, do is the 4th Number to the content in eo
5 ! ; ey eAle S&:..

h

, 000 for Ales

Gallons.

CHAP,.

7 7s

gt Bér. Samnitel Eofter his

oo CHAP. XII.

‘Of the two Scales upon the inner edge of the other Leg, which are
dsvined the one into 14, the other into 20 equal parts.

Hele two Scales ferve further for Gauging: of Veffels by the
J. mean Diameter, and the ufe of them‘is.

To find fuch a mean Diameter between the D&meters of. the Ss
headand Bung, as fhall reduce the Vefleltoanequal Cylinder
of the fame length with the Veffel. .

‘This way of Gauging is in ufe, aad for our common Veflels may
ferve as coming fomewhat near the eruth, :

Having meafured, either in Feec or Inches, the two Diametersat
Head and Bung, take their difference and count it npon the Scale of a
20. And taking that length from thence, and applying ic cothe Scale
of 14, feewhat Number it cutsthere. Addcthat Number tothe leffer
“Diameter, which is the Diameter at the Head, and thefumwillbe
the mean Diameter. : 4

To Gauge by the mean Diameter.

You muft firft meafure the Dimenfions of the Veffel with a Scale |
of Foot or Inch meafure. | : py
Then take the Number of the mean Diameter out of the repre- #
fentative fae. Sand meafure it upon the Line of Superficies M
( effeeming che faid whole Line but 10 ) and keep the Number thatic
falls upon. Then upon the Line of Numbers fay, Astetothe Number |
kept ; fo ee length in aes oto a fourth. : |
Then if the length be ier ib Inches, and the former work were
Berineh 7-49 tor Wine 9.35 forWine, —
alfo for inches,fay,as 8 tof ‘fuente ha Ror a5 10 t03 9:35 {OF pi: ;
_ Sethe former fourth Number to the Number of Gallons in the Vela.

Or

~~ - pt al Veg, bonnes a a » y Se a
, : - > ve
G . 4 i]

alteration of the Sector. 193
Or if the length were given in foot meafure, and the formar work

: al
were alfo for Feer, fay, As 1 tos i : aa re reid 2 fo the former
be 9

fourth Number tothe contentin Gallons. :
Or elfe take the Veffels length out of the reprefenratives i ie ¢
2 3

and epply them to the > ie ¢ meafure Scales, and obferve whae

Numbers they there fall upon. Then fay, as $10 this new length ;(0 the
Number kepr before, to che Number Spas 2 Gallons in the

Veffel. |
This laft way isperformed by one work upon the Line of Numbers,

wiereas the other requires two.

CHAP. XIV.

_ How to meafure Cartridges of Ganpowder to know how many pound
are contained in them.

1.1f the Cartridge be of 4 Cylinderical form.

Irft meafure the Diameter and length of ic witha Scale of Deci-
mal inches. Then count che Diamecers length. upon the Line of
Lines, ( counting che whole Line of Lines as 10 reprefentative inches)
| amd wich your Compafles, cake the Jength of char number from the
Line of Lines, and epply itto the Line of Superficies, ( which now
in this work muft be fuppofed to contain 10 parts) and nore the Num-
_ ber which it reacheth unto. Thenupon the Line of Numbers, fay,
_ As 4o% is to that Number noted, fo the length of the Cylinder, tothe
| Number of the pounds of Gunpowder.

2.Jf the Cartridge be of a Conical form.

___ Meafure the Diameter of the Bafe, as before, and the length of the
Cone likewife, both with a Scale of Decimal Inches, and count the
_ Diameter upon the Line of Lines, and apply itto the Line of Super-

5 Cc — ficies,

\

| 194 ws Mr, Samuel -Fofter has -

ficies, noting the Number thereon, as was done before. Then fay, ba

As 121 #tothe Number before noted : So the length,toche ccmient in
pounds of powder. See

~

2 If the Cartridge be a vefetted Coney

Meafure the Diameters of both Bafes, andthe length by aScale of
Decimal Inches: Then count the greater Diameter upon the Lineof ~_

Lines, and meafure it upon the Line of Superficies, noting the Nam-

ber, as was done inthetwo former ways. Afterwards, upon the Line

of Numbers, fay ; Asche greater Diameter to the leffer ; So the no-

ted Number to a fecond,and fo that fecond coathird. Add thefe three ~

the firft, fecond and third Numbers together, and keep the fum. Then

- fay again, As 121 z isto the faid fum; Soisthelength, cochecontent

x) 2]
in pounds of powder.

Or you m1y count both the Diameters upon the Line of Lines, and
transfer them both to the Line of Superficies, and note what two
Numbers they cut, count the fame two Numbers upon the Line of
Nambers, and bifec&t che diftance between them, fo fhall you finda

middle Number, which, with che two former, will make up three

Numbers, che fame which were found, the other way. Then (as be-

fore ) add thefe chree Numbers together, aud keep che fum, and again —

fay, As 121 7 isto the faid fum;So is the lengthto che content in pounds
of powder.

The End of Mr. Fofters alteration of the Seéfor. ;

My oe Se tee A oy atte’ Se" jell femmemouy Wie Coe ae Se sy et eRe Ke Po ae! Ca ogee
- ? - 4 ‘ 5 : bd q e ta
% f *
'

Belles nk eOnteript,

hs Pon the Scheme of this Sector, as Mr. Fofter hath contrived
Sb there are fome other Lines inferted iw the pare places

| thereof, which do not go up tothe Center: As, Firft, Line of

three hours, with their halves and quarters, which Line és noted at

| every whole hour with %, aud at every half and quarter, with «

_ bistle Line thus\, The ufe of this Line is chiefly in Dialling, and

| the manner of ufing it w fafficiently fhewed in other of his works,

and it moft exceliently andexpedicionfly performeth that manner

of Dialling, which Adr, Gunter teacheh at the end of his third

Book of the Sector.

| There ave alfo other Scales, as one of Metals, and another of

| Segments of 4 Circle, the ufes whereof are the fame, as Mr, Gune-

ter hath fhewed at large: and there ts alfe added another

|

|

| Line by Adr. Fokter, which t alfo called a Line of Segments ; that
_ of Mr, Gunter reprefenting the Segments of a Circle + the other
of Mr, Fofter, the Segments of a Sphere, and hath like ufe in
| Spheres, as the other hath in Circles,

Vale.

ONE

pane a ae ”
ow Sint \ Sige rite che
q : - yes, eN"
“RD aE

% ah bs tice

s

« %

tai

tee ne
art
et

wy 4
‘ ¢* hs *

et tl

: oy: a i 4? , 4 Y ; 4 ‘ay
| Mi & |
ssp oe thorn i we Sates Nair sera eT re aoe ae’ «cep ts be
Va as z - i] = a
} . : 7 eet 3 >. ‘ Lies

baAtp

THE ai
CROSSSTAFF;

iP hreesb 0 '0'Ks):

: The Firft,
Containing its Defcription, and the Ue thereof in .

| taking of Heights and Diftances,

The Second,

Contains the Ufe of the Lines thereon in the Meafuring ©
of all manner of Superficies, and Solids, as Board,
Glafs, Land, Timber, Stone, and Gauging of Velflels, .
as alfo inthe famous Art of NAVIGATION.

The Third,

Contains the Ufe of the Lines of Numbers, Sines,
and Tangents in Dialling, an Excellent and
Compendious Treatise, fully teaching, and —
amply explaining the Grounds and Reafons thereof,
from a Projeétion of the Sphere ia Plano,

To which is added,

An APPENDIX, containing the Delcription and -
Ufe of a fmall Portable Quadrant, for the more eafie
finding the Hour and Azimuth, and other Solar .
Conclufions of more frequent Ufe. |

By Edmund Gunter.

LONDON,

Printed by 4. C. for Francis Eglesfield, and are to be fold at -
. the Marigoldin St Pasls Church-yard. 1673.

a i on P-

at lon ane Bs bis. aaliqinaty i hau}
esandl aa: bas sy isk 19 gaia

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poet hivaoM ay hi agai il 2 Pant
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th bn Vio snigusd ba Kp 2: 7 adm h ne

| ADIT AD rv AVE ta HA anomie oils ni ols
nae y Ne teas bid? et rT ‘ :

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—

: a ea earch a Ky ganic ocd. to otis Swi Ue

| ing ‘4Susl llooxd ne gaillsiCl ob 2jasgec ld pas §
ee hes | yaid3ses. yligl art Aw az a canada 7g

i Aete rane anoineA bas shrvorwd: ol) onintsiqzs vq: “fi
i Ape 27 ee Sak ose q eo iy Aone nS a i mote

= WODDS ae isidw of. Re

(ae poidaO $d gaigisyno> kid is a <4 Ai 4
ghar. sx0m ada rot aasronyO oldsiy Rat Ey to Te ie
ep Se 13 ang bas Adipensd bhe wolt te ‘aad a fie
“ue | ie ah inure eel snaupait 21018 j03 a oistoaoeyal |
Sas? ec Iie hcink ip ak it et pea Tas <

as. _weQ Twat a
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als Voss espdintenich Pinu t @ i BgtM

FIRST BOOK
CROSSSTAFF.
- CHAP, 1. stor th

Of the Defeription of the Staff. .

He Crofs-Staff, is an Inftrument well known to our:
Seamen, aod much ufed by theanciénr Aitrono-
mers and others, ferving Aitronomically for ob-
fervation of the Alticuceand Angles of diftance-
inthe Heavens; Geometrically for Perpéndi-
cnlar Heights and Diftances: on Land and -
Sea. babe

3 The Defcription and feveral Ufesof it areex..

‘tant in Print, by.Gemma Prifius in Litine, in Englith by D. Hood. L-
differ fomething fromthem both, in the Projection of chis Staff, bur

_ fo as their Rules may be applyed unto it, and: all their Propofitions be

- wroughe by ic ;. and therefore referring che Reader to their Books, I

_ fhall be brief in the Explanation of that which may be applied from.

theirs unto mine, and focometathe Ufeof thofe Lines which ‘ere oF.
my Addition, noc extantheretofore. pa

- The neceftary partsof thie Inirument are Five; (s.) The Sat
(2.) The Crof, And(3.Jchethree Sights; The Stath which Imade

for my own ufe, is a full Yard in length, that fo ic may ferve for
The -

meafure.

200 The Defcription of Lines. : |
-_ The Crofs belonging to it is 26 Inches } becween the two oftward
fights, If any would have it ina greater form, the proportion between —
the Staff and the Crofs, may be fuch as 3 Go unto 262,

_ The Lines infcribed on che Staff are of four forts: One of them
| ferves for Meafure and Protradtion : One for obfervation of Angles: —
One forthe Sea-Chart ; andthe four other for working of Proporti-

ons in feveral kinds. Jj 3

The Lineof Meafure is an Inch Line, and may be known by his —
equal quarts, the whole Yard being divided equally into 36 Inches,
and each Inch fubdivided firft into ten parts, and cheneach centh parc
into halfs, :

The Line for obfervation of Angles may be known by the double
Numbers, fet on both fides of che Line, beginning at the fide at 20
and ending at90: on the other fide at 40, and ending at 180: and
chis being divided accordingto che degrees of aquadrant, I callit che

Tangent Line on the Staff.

The next Line is the Meridian of a Sea-chart, according to Aterca-
tors Projection from the Equino€tial to 58 gr.of Latitude, and may be
known by che letter 44, and the Numbers r, 2, 3, 4, unto 58. |

The Lines for working of proportions may be known by their un-
equal divifions, and she numbers at the end of each Line. ‘3

1. The Line of numbers noted with the Lecter N, divided unequal-
ly into £000 parts, and numbred with 1, 2, 3,4, unto10.

2. The Line of Artificial Tangents is noced with the letter T, divi-
ded unequally into 45 degrees, and numbred both ways, for tae Tan-
gent and the Coraplemenr. ' |

3. The Line of Artificial Sines noted with the letter S, divided un-
equally into 90 degrees, and numbred with 1, 2, 354, un go.

4. The Line of Verfed Sines for more eafie finding the hour and
Azimuth, noted with V, divided unequally into abour 164. gr. 50 m.
numbred backward with 10,20, 30, unto164,. 000 tf

Thus there are feven Lines infcribed on the Scaff: there are Five _
Lines more infcribed on the Crofts. B

i. ATangenc Line of 36 gr. 3 m. numbred by 5, 10, 1§, unto 355
the midft whereof is at 20 gr. and therefore I call ic the Tangent of
205 and this hath refpeé unto 20 gr. inthe Tangent onthe Staff.

2. A Tangent Line of 49 gr. Gm. numbred by 5, 10,1 5,Unto 455
the midft whereof isat 30 gr. and hath refpe& unto 3ogr. inthe Tan-
gent on the Staff, whereupon J call itthe Tarigent of 30. sh )

ce

3. A Line of Inches numbred with I, 2, 3, unto 263 each Inch

‘equally fubdivided into ten parts, anfwerable co che Inch Line upon

the Staff. Pe ine
- 4. ALine of feveral Chords, one anfwerable toa Circle of twelve

Inches Semidiameter, numbred with 10, 20, 30, unto 60, another a
Semidiameter of a Circle of fix Inches; and the third to a Semidia-
mecer of a Circle of three Inches, both numbred with 10, 20) 30; un=
£0 90.

s- Acontinuation of the Meridian Line from 57 gr, of Latitude
unto 76 gr. and from 76 to 8 4 gr.

Por the Iafcription of thefe Lines. The firft for meafure is equally
divided into Inches, and tenth parts of Inches. yas

The Tangent onthe Staft for obfervation of Angles, wich che Tan-
gent of 2oandthe Tangent of 30%onthe Crofs, may all three be in-
fcribed out of the ordinary Table of Tangents.

The Staff being 36 Inchesin length; the Radius for the Tangent on
the Staff will be 13 Inches and 103 partsof 1000: fo the whole Line
will be a Tangent of 70 gr. and muft be numbred by their Comple-
ments, and the double of their Complements, the Tangent of 10 gr,
being numbred with 80 and 160.

The Radius for the Tangent of 20 on the Crofs, will be 36 Inches,

and the whole Line between the Sights a Tangent of 36 gr.3 m.accord-.

_. ing as it is numbred, The Radius for the Tangent ot 30¢r. on the
— Crofs, will be 22 inches and 695 parts of 1000: fo the whole Line bee
eween the fights will contain a Tangent of 49 gr. 6m. infuch fort as
they are numbred. |

The Meridian Line may be infcribed out of the Table which I fee
down for this purpofe inche ufe of che Sector.

The Line of Numbers may be infcribed out of the firft Chiliad of
Mr, Briggs Logarithms : and the reft of the Lines of proportion oue
__of my Canon of Artificial Sines and Tangents, and in recompence
thereof this Book will ferve asa Comment to explain the ufe of my

Canon. |

Dide i-\utnaty GEA Bo

| The Defeription of Lines. 205

BCR S
F a

non sts The fe off the Lines of Inches

~ ce SIE EES OE . me . ; : :

CHAP. IL

rhe ufeof the Limes of Inches for perpendicular heights ana
diftances.

N taking of heights and diftances, the Staff may be heldin fuch

fort, that itmay be even with che diftance, and the Crofs parallel
with the height : andthenif the eye atthe beginning of che Staff fhall:
fee bis marks by che inward fides of the two firtt fights, chere will be
fuch proportion between the diftance and the height, as is between the
parts intercepted on the Scaff andthe Crofs. Which may be farther
explained in ehefe Propofitions. . :

%e
a
®
‘eo
es
e

+ e? "eee,
B Mesaaonn gogPasase Sgneausinas '*_or ‘auner kl
Hee D

1, Tofind an height at one feation, by knowing the diftance.

Set the middle fight uato the diftance upon the Staff, the height mill
-be found upon the Crofs. For, | !

Asthe Segment of the Staff

unto the Segment on the Crofs s
‘So is the diftance given,

unto the height.

Cee WORT, Gl bee ree Ne eT ONCE Pe BD OR Ore ee gate a Be oe eae Pe
Need : Wes) Cae oe Stag) Moo. .

-

for heights and diftences, has he ews

As if the diftance A B being known to be 256 feet, ie were required
tofind the height BC: firt I place the middle fighe at 25 inches and
6 partsof 16; thenholding the Staff level with the diftaace, I raile
the Crofs Parallel unrothe height, in fuch fort, as that my eye may fee _-
from A the beginning of the inches on the Staff by the fight E, at the
beginning of the inches onthe Crofs unto the mark C: which being
done; if I find 19 inches and 2 parts of 10 intercepted on the Crofs
ih the fights at Eand D, I would fay the height BC were 192
eet.

Or if the obfervation were to be made before the diftance were
meafured, I would fer the middle fight either unto 10 inches, or 12 or
16, Or 20, or 24, or fome fuch other number as might beft be divided
into feyeral parts, and then work by proportion. As if in che former °
examplethe middle fight were at 24.0n the Staff, and 18 onthe Crofs,

. it fhould feem that the height is j of the diftance 5 and therefore the

diftance being 256, the height fhould be 192.
. 2. Tofind an height by knowing fome part of the fame height.

Asif the heighe from G to C were known to be 48, and it were re-
quired to find the whole height BC, either put che third fight, or
fome other running fight upon the Crofs between the eye and the mark

G. For then,

As the difference between the fights Was
unto the whole Segment of the Crofs :

So isthe part of the height given,
unto the whole height.

If chen the difference between the fights at Eand F, hall be 45 ,snd
the Segment of the Crofs ED 180, the whole height BC will be

found to ber g 2.

3. To find an height at two ftations, by knowiug the difference of the fame
tations.

As the difference of Segments onthe Staff,
unto the difterence ef {tations ;
So is the Segment of the Crofs, ‘
unto the height. Dd 2 Suppofe

“

t

204 sss he wfe of the Lines of Inches

t

. Suppofe the firt ftacion being at H, the Segment of the Crofs E D

were 180, andthe Segmentof the Staff H D 300: then coming 64. "
feet nearer unto B, in a dire& Line unto a fecond ftationat A, and —

making another obfervation, fuppofe the Segment of the crofs E D>
were 180 as before, andthe Segment of the Scaff AD 2405 take 240 —
out of 300, the difference of Segments will be 60 parts. And

As 60 parts unto 64 the difference of ftations: —
So D E 180 unto BC 92 the height required,

In thefe chree Propofitions there is a regard to be had of the heighe
of the eye.For the heighe meafured,is no more then from che level of
the eye upwards.

4. To find adiftance, by knowing the height.

As the Segment of the Crofs,
unto the Segment of che Scaff :
So is the height given,
unto the diftance.

So the Segment ED being 18, and DA 24, the height C B 1925
will thew the diftance A B to be 256. | |

5. Tofind adiftance, by knowing part of the height. .

As the difference berween the fights,
unto the Segment of the Staff :

So is the part of the height given,
unto the diftance.

And thus the difference between E and F being 45, and che Sep.
ment D A 240; the part of the height G C 48, will give the diftance
A Bro be256.

6. To i h diftance at two ations, by knowing the difference of the fame

Ghi0ns, +

Asthe difference of Segments on the Staff, -
untothe difference of ftations < So

|

| breadth BC, the proportion wall hold ;

1
F
|

for heights and difances, | 109

a | So is the whole Segment,

unto the diftance...

And thus the Segment of the Crofs being 180, the Segment of the
Staff at the firft tation 240; atthe fecond 300, the difference of the
Segments 60, and the difference of {tations 64, the diftance AB at.
the firft ftation will be found tobe 256, and the diftance H Bat the {e-.
cond ftation 330. | *

7. Tofind a breadth, by knowing the diftance perpendicular to the breadth,

". This is all one with the firt Propofition. For shis breadth is.but.an:

height turned fideways ; and therefore

As the Segment of the Staff,

unto the Segment of che Crofs.; .
So is the diftance :

unto the breadth..

And thus the Segment of the Staff being 24, and the Segment of ©
of the Crofs 18, the diftance A B 25.6, will give the breadth B C to be
192. .

~ 8: To fisd a breadth at two ftationsin a Line Perpendicular tothe breadth,
by knowing the difference of the fame frations..

This is alfo the fame with the third Prop, and therefore
As the difference of Sesments on the Staff,
unto the difference of ftations:
So the Segment on the Crofs between the two fights.
unto the breadth required.

And thus the difference between the {tations at A and H being 64,

| the difference of Segments on the Staff 60, the Sep mene of the Crofs
180, the breadth B C will be found tobe 193.

In like manner may we find the breadth G C, for having found the »

Ag.

We Ue hg Pa a ae

or
}

Ha6 Of taking bntatss

As DE is unto F E, fo BC unto G C, Or otherwife,
As H 4untoH A, fo F Eunto GC. ,

Neither is i¢ material whether the two ftations be chofen at one end
of the breadeh propofed, or without it, or within it, if the Line be-
eween the ftations be Perpendicular unto the breadth s as may appear,
if in ftead of the ftations at Aand H, we make choice of the like (ta-
tions at] and K, 7

‘There might be otber ways propofed to work thefe Propolitions,by
holding the Crofs even with the diftance, and she Scaff parallel wich the _
height: but thefe would prove more troublefome,and thofe which are —
delivered are fufficient, and che fame with thofe which others have fet
‘ | flown under the name of the Facebs Staff.

\

Pg
te

oo.
aes

cf

| og FX
a —————

The wfe. of the Tangent Lines.

CHAP. MUI,

_ (ee
A ae =e SS SSS | ees a SS PS iN
% : i :
* had
A, t jg &
¢ -“ 5 7
% 4 af
! % = rt
A =
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| 3 = se
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e ay rt >
% ; : :
% > s } e
° 7 ,
Any 4 3 ¥ = .
WG + : i fff
bit ° © s He!
AREAS, REE LODE Cs SRA US
0 i. es eee ee eee ee eae
% . ‘ * ie f > ue
% % ° he $ ©
& 2 saogennner® H "Es. ofa, = £
% yeast - Or : Jo" peer, &
% xo ¢ RON,
% 987 . s e Teo, Sy
& ° eo 2 wy /
ys. eG F Fol,
a om . a a s ,
a “ e, oe 2 a - 7
ay? ck A : , YO%,-
A) % s Ss : 2 £ %
LS & 3 ~4 © D ¢
bass % 2 S : 2 = =
ee) » & BE pe 5
s /* % 3 i e * @
2 3 ° s - a ¢
x AA i ie e O
= % é s if ~ ?
<Q + - : Peay
2 +, . ~ rhe iY
e ° ° . ® SJ
nS) EA a Oe D ,
-—ot » ~ e . ° @
=. % on 4 e @ e
rf SS Se eS
re * Sik aS
§ ee J
& @ ee fs ry
i Sais 2 2
° + Scira ? 2
3 oe o@ © ]-M e y
eS i | Pr
- 3 os th- me
Cy cx» , =
e » As He
4 SHG
Ef AX;
(
as

207

| The He of the Tangent Eines in taking of Angles.

1. To find an Angle bythe Tangent onthe Staff. .

Et the middle fight be always fet to the middle of the Crofs, no- -
Lies with 20 and 30, and then the Crofs drawn nearer the eye,

wocill ghe marks may

be feenclofe withinthe fights, For feif the eye

at...

The ufe of the Tangent Liness “a
at A (that end of the Staff which is noted with 90 and 180.) beholding
-the mark K and N_ between the two firft fights, C and B, or the.

marks K and P becween the ¢wo outward fights, the Crofs being drawn.
down unto H, fhall ftand ar30 and 60, inthe Tangent onthe Staff : ie
‘ftheweththe Angle K ANis 30gr. the Adgle K AP 60 gr. theone
doubleto the other; which isthe reafon of the double-numberson —
this Line of the Staff: and chis way will ferve for any Angle from
20 gr. toward 90 gr or from 40 gr. coward 180 gr, Butif the Angle
be lefsthan 20 gr. we muft then make ufe of the Tangent uponthe —
Crofs. | |

208

‘

2, To fad an Angle by the Tangent of 20 upon the Croft.

Set 20 unto 20, that is, the middle fight to the midftof the Crofsat _

the end of the Staff, noted with 20, fo the eye at A, beholding the

marks Land N, clofe between the two firft fights, C and B, thall fee
shem in an Angle of 20 gr.

If che marks fhall be nearer together, as are M and. N, then draw
‘inthe Crofs from C unto E: if they be farther afunder, as are K and |
N, then draw out the Crofs from C unto F; fo the quantity of the
Angle fhall ftill be found in the Crofs in the Tangent of 20¢r. atthe
end of the Staff: and this will ferve for any Angle from 20 cowards
35 er.

3. Tofied an Angle by the Tangent of 30 pon the Crofs,

This Tangent of 30 is here put the rather, thatthe end of the Staff —
refting arthe eye, the hand may more eafily remove the Crofs : forie
fuppofech the Radiusto bene longer than AH, which is fromthe eye —
atthe end of theSeaff unto'30 gr. about 22 inches and 7 parts. Where- ~
fore here fet the middle fight unto 30 gr. onthe Staff, and then either
draw the Crofs in or our, untill che marks be feen between che two
firft lights ; fo the quantity of the Angle will be found in the Tangent
of 20, which is here reprefented by che Line GH; and chis willferve _
for any Angle from o gr. coward 48 gr. 7

4. Toobferve the Altitude of the Sun backward.

Here it is fit to have an horizontal fight fetto the beginning of ihe
: Sta >”

The ufe ofthe Tangent Lines; = ° 209
‘Staff, and chen may-yon turn your back toward the Sun, and your
Crofs toward your eye. If the Altitude be under 45 gr. fet the middle

fight to 30 onthe Staff, and look by the middle fight through the Ho-
“rizontal unto the Horizon, moving the Crofs upward or downward,

untill the upper fight do thadow the upper half of the horizontal
fight: fo the Altitude will be found inthe Tangent of 30,

. Tf the Altitude thall be more than 45 or. fee the middle fishe unto
the midft of the Crofs, and look by the inward edge of che lower
fight through the Horizontal tothe Horizon, moving the middle fighe
in or out, untill the upper fight do fhadow the upper half of the Ho-

-between4.0 and 180, L |

-rizontal fight + fo the Altitude will be found inthe degrees on the Staff

5. To fet the Staff to any Angle given.

This isthe converfe of the former Propofition: For if the middle
fight be fet to his place and deg ree, the eye looking clofe by the fights
as before, cannot but fee his obje& in the Angle given.

6. To obferve the Altitude of the Sun another way.

Set the middle fight to the middle of the Crofs, and hold the Hori-
zontal fight downward, fo as the Crofs may be paraflel coche Hori-
zon,then isthe Staff Vertical; and if the outward fights of the Crofs
do thadow the Horizontal fight: the Complement of the Altitude will
be found inthe Tangent on the Staff.

7.10 obferve an Altitude by Thread and Plummet.

Let the middle fight be fee to the midft of the Crofs,and to that end
of the Staff which is noted with 90 and 180 ; then having a Thread and
a Plummet at the beginning of the Crofs, and turning the Crof up-
ward, and the Staff cowards the Sun, the Thread will fall on the Com-
plement of the Altitude above the Horizon, And this may be applied
to other purpofes. | |

8.7 0 apply the Lines of Inches to the taking of Angles.

Bane Angles be obferved between the two frit fights, there will
I + i Ee be

7

210° The ufe of the Line of Chords.

be fuch proportion between the parts of the Staff and the parts of the
Crofs, as between the Radius and the Tangent of the Angles

Asif the parts intercepted onthe Staff were 20 inches, che parts on
the Crofs 9 inches, Thenby proportion as 20 unto 9, fo 1e000 unto
45000 the Tangent of 24 gr. 14m.

Bur if the Angle fhall be obferved between che two outward fights,
the parts being 20 and.o as before, che Angle will be 48 gr. 28 m.
double unto the for mer, : |

In all chefe there isa regard to be had to the Parallax of the eye,and
~ his height above the Horizonin obfervations at Sea; ro the Semidia-
meter of the Sun, his parallax and refraGtion,as in the ufe of other flaves.
And fo this will be as much, or more than that which hath been here-
- sofore performed by the Crofs-Staff.

=

CHAP, Ivy. )

The ufe of the Lines of equal abd joyned with the Lines of
| | Chords.

2 ks Lines of equal parts do ferve alfo for protraétion, as may ap-
pear by the former Diagrams, but being joyned with the Lines of
Chords, which I place upon one fide of the Crofs, they will farther
ferve for the protraétion and refolucion of right Lined Triangles;
whereot I will give one example in finding of a diftance at two {tations
otherwife than inthe Second Chapter.

Let the diftance required be AB, st A thefirft tation, I make
choice of a ftation Line towards C,and obferve theAngle B A C by the
Tangent Lines, which may be 43 gr. 20m. then having gone an hun-
dred paces towards C, I make my second {tation at D, where fuppofe
I find the Angle B DC to be 58 gr. or the Angle BD Atobe
this being done, I may find tie diftance A Bin this manner.

1. Idrawaright Line A C, reprefenting the ftation Line:

2. [take 100 out of the Lines of equal parts, and prick them down —
from A the firft ftation unto Dehefecond.

3. Topen my Compaflesto one of the Chords of 60 gr.and fetting -
ene footinthe point A, with the other I defcribean occule Ark of a
Circle tnterfecting the {lation Line in E, i |

‘I12gr

4-7.

The wfe of the Lines of Chords. att

4. I take out of the fame Line of Chords a Chord of 43 gr. 20°
( becaufefuch wasthe Angle at the firft ftation ) and this I inicribe into
that occule Ark from EuntoF, which makes the Angle FA D equal
to che Angle obferved at the ficit tation. haat

5 I defcribe another like Arkuponthe Center D, and infcribe ins
to it a Chord of §3 gr. from C unto G, and draw the right Line D G,
which doth meet with che other Line AF in the Point B, and makes
the Angle B DC equal to the Angle obferved at the fecond ftation.
Sothe Angles in the Diagram being equal tothe Angles in the field
their fides will be alfo proportional ; and therefore, —

6. I take out the Line A B with thy Compaffer, and meafuring it in

__ the fame Line of equal parts, from which I took AD, I find it to be

335, and fuch is the diftance required.

Ee2 CHAP.

ie fe"

21Z

The ufe of the Meridian Line,

1.” 7 He Meridian Line, noted withthe letter M, may ferve for the

more eafie divifion of the plain Sea-chare, according to Mer-
cators Projection, For if you fhall draw parallel Meridians,each degree
being half an inch diftant from other, the depree of this Meridian

Line on the Staff fhall give the like degrees for the Meridians on the
Chart, from the Equinoctial toward the Pole: and then if through

chefe degrees you draw ftraight Lines Perpendicular to the Meridians,
they fhall be Parallels of Laticude:

If any delire to have the degrees of his Chart larger than thofe
which I have put on the Staff, he may take thefe and increafechem ina
double, or treble, or a decuple proportion at his pleafure.

2, This Meridian Line being joyned with the Line of Chords, may
ferve for the protraction and refolution of fuch right Line Triangles .
as concern Latitude, Longitude, Rumb and Diftance inthe practice of
Navigation. Astay appear by this example.

Sappofe two places given, A inthe Latitude of 50 gr. D inthe La-

situde of g2¢r. 2, the difference of Longitude between them being

6 gr. and.tec it be required to know, firft, what Rumb leadeth from
the one place tothe orher; fecondiy, howmany degrees diftant they
are afunder.

:. Idraw aright Line A E, reprefenting the Parallel of-the place
from whence I depare. ang |
. 2,-Ieske 6 gr. for the difference of Longitude, eisher ont of the
Line of Inches, allowing half aninch for every degree, or out of the
pegionins a ae eHslen ne $a there the Meridian degrees dif-
ervery lictle from the Equino@tal deprees ) and thefe6 er. {
down in the Parallel aa Ato E. Bee sam
3. InAand E, ereét two Perpendiculare,
the Meridians of both places, -.

» “4 Ftake the difference of the Latitude from 5@ gr. tO §2e7. 30m,

Oug ei

A Mand ED reprefenting i

| The afe of the Meridian Line. Eo
' out-of the Meridian Line, and prick it down in the Meridians from A
unto M, and from Eto D, and draw the right Line M D for the Pa--
- yallel of the fecond place, and the right Line A D for the Line of di-

ftance between both places: focthe Angle M AD fhall give the Rumb »
that Jeadeth from the one place to the other.

,

ay
TTI} Att ot

i moo ri

53
a a ST

|

oO
* Pi
TN

2

5.To find the quantities of this Angle M A D,I may either make ufe-
~ of she Protraétor,or elfe of a Line of Chords,and fo lopenmy Com~
~ paffes unto one of the Chords of 60 gr. and fetting one foot in the:
Point A, withthe other I defcribe an occule Ark of a Circle, inter-:
- fedting the Meridian inF, and the Line of diftance inG; then I take
the Chord of F Gwith my Compaffes, and meafuring ic in the fame-
Line of Chords as before, I find it 6 gr. 4 : and fuch ts the Iaclinati-
tion of the Rumb to the Meridian, which is che firft thing that was rez»

uiredé
: | | | 6. To

ag ee |

214. The ufe of the Meridian Link

6. To find the quantity of the Line of diftanceA D, I take it oue

‘with my Compafies, and meafuring itin the Meridian Line, fetcing one
foot beneath the leffer Laticude,and the other foot as much above the

greater Latitude, J find about 4¢r. + intercepted between both feet :

~.-and fuch isthe diftance upomthe Rumb, which is the fecond thing thae

Was required. |

But if this example were protracted according to the common
Sea-Chart,; where the degrees of the Equinoétia! and Meridi-
an are both alike; the Rumb M AD would be found to be

above 67 gr. 23 m. and AD the diftance upon the Rumb about

6 gr. 4.

Stppofe farther, that having fee forth from A toward D, upon
the former Rumb of 56 gr. 15 wm. NEGDE, after the Ship had
run 36 Leagues, the wind changing, it ran 50 Leagues mere upon the
feventh Rumb of E/N, whofe inclination to the Meridian is 78 gre
45 m. And let it be required to know what Longitude and Lati-
tude the Ship is in, by pricking down the way thereof upon the
Chart.

Havind drawn a blank Chare as before,with Meridians and Parallel,
according to the Latitude of the places propofed. |

1. T would make an Angle M A Dof 56 gr. 15 m. for the Rumbof _
NEDE, which is done after this manner: I open my Compaffes to
one of the Chords of 60 gr. and fetting one foot in- the Point A,with
the other I defcribe aw occult Ark of a Circle, interfecting the Meri-
dian inF : thenI take 56 gr. 15m. out of the fame Line of Chords,
and prick them down from FuntoG. focheright Line AG fhallbe
the Rumb of NEDE. 7 }

2. I would take 36 Leagues out of the Meridian Line, extending
my Compaffes from 50 gr. t0§ 1,48. or rather from as much be-

_ low $0 as above §1, and prick chem down upon the Rumb from A un-
tol; fo the Point I fhall reprefent the place wherein the Ship was

when the wind changed. And this is in the Latitude of §t gr. om
and in the Longitude of 2 gr. 21 m. Eaftward from the Meridian
AM, hd | “pi
3. By the fame reafon, Jmay drawthe right Line] K fer the Rumb _
of EDN, and prick downthe diftance of 50 Leagues from J untoK :
fothePoint K fhall reprefent the place whither the Ship came, after
che running of thefe 50 Leagues: and this is inthe Latitude of 51 gre”
30 m, andin Longitude 6 gr, 16%. Eaftward from the firft Meri-
dian

a The ufe of the Meridian Line. ‘2re
dian AM; and therefore 16. Eaftward from the fecond Meridian
ED. eA
But if thefe two courfes. were to be pricked down by the com-
mon Sea-Chart, the Point I would fallin the Latitude of §1 gr. om,
and the Point K in the Latitude of g1gr.30. But the Longitude
of I would be only 1 gr. 30 m. and the Longitude of K only
3 gr. 57 m. More: both chefe do make but 5 gr: 27 m. for the
difference of Longitude between the firft Meridian A M, and the
Point K: whereby it fhould feem thar che Point K is yee 33 m..°
Weftward from the Meridian of the plice to which che Ship was.

bound.
Such is the difference between both thefe Charts.

CH AP;

si ae, a

te

cet a

b eu AGRE LY p *

CHAP, VI.
(The wfe of the Line of Numbers

He Line of Numbers here noted with 1,2,314,
unto 10, is compleat in thole divifions which
are between 1 and 10: the-other like divifions at
the beginning of the Line do ferve rather to an-
{wero the firft degrees of the two other Lines of |
Sines and Tangents, than for any neceffity, which |
is the caufe why fome of them are omitted. And
here, as in che nfe of other Scales, the figures
1, 2,3,4.that are fet down upon the Line, do fome-
times lignifie themfelves alone, fometimes 10, 20,
30, 40, fometimes 106, 200, 309, 4.00, and fo for= _
L 49 ward, asthe matter fhall require. The firft figure
of every Number is always that which ts here fet
-44°down, the -reft mutt be fupplied according co che
nature of the queftion. im.

£

1. Having two Nunsbers given, to find a third in come
1 13 tinwal-propcrtion,a fourth, a fifth, and fo forwards —

m2 Extend che Compaffes from the firft Number —
1$ unto the fecond; ther may you turnthem fromthe |
fecond to the third, and fromthe third to the ©
+46 fourth, and fo forward. the
Let the two numbers given be 2 and 4, extend
‘7 the Compafles from 2 to 4, then may you turn |
them from 4.to 8, and from 8 to 16, and from 16 —
to 32, and from 32 to 64, and from64t0 128.
Or if one foot of the Compaffes being fet to 649 |
| the other fall out of the Line, you may fet ito ©
}-mnw another 64 nearer the beginning of che Line, —
and there the other foot will reach to 128, and |
from 128 you may turnthem to 256, and fofor
ward. ,

1 EP tee ty Seed oY Seen an NAME ES Pen Br Ere
" th)

mae nO e

The ufe of the Line of Numbers, 217
Or if the two firft Numbers given were 10 and 9 : extend the Com”
__ Paffes frow ro atthe end of the Line, back unto 9, then may yoyturn
___ them from’g unto 8, 1, and from 8, 1, unto 7,29. And fo if che two
_ firft Numbers given were r and 9, the third would be foundto be
81, the fourth 729, withthe fame extent of the Compaffes.

In the fame manner, if the two firft Numbers were 10 and 12, you
may find the third proportional to be 14,4, the fourth 17,28, And
with the fame extent of the Compafles, if che ewo firft Numbers were
I uh 12, the third would be found to be 144, and the fourth to be
1720,

*"-. > * ee gee Sah ew OD OY OME a Ae
eS een army Emre SP ey F ty ere

2. Having two extreme Numbers given, to find a mean proportional
between thenp.

Divide the fpace between the extreme Numbers into two equal
parts, and the foot of the Compaffes will ftay atthe mean propor-
tional. Sothe extreme Numbers given being 8 and 32, che mean be-
tween them will be found to be 16, which may be proved by the
former Propofition, where is was (hewed, that as 8to 16, foare16
£32.. =”

3. Tofind the {qaare Root of any Number given.

The fquare Root is always the mean proportionl! between 1 and
_ the number given, and therefore to be found by dividing the fpace be-
tween them into two equal parts; So the Root of 9 is 3, and the ©
Root of 81is9, and the Root of 144 is 12, and the Root of 1440
- almoft 38. : :
If you fuppofe Pricks under tae Number given, (asin Arithmetical
extraction ) and the laft Prick to the lefe hand (hall fall under che lat
| figure, which willbe asoft astherebe odd figures, the unity will be
belt placed at 1 inthe middle of the Line: fo the Root and the Square
will both fall forward toward the end of the Line. But if che lait
Prick fhall fall under che laft figure but one, which will be as oft as
there be evenfigures, then the unity may be placed at 1 in the begin-
ning of the Line, andthe Square in the fecond length, or rather che
Unity may be placed at 10 in the end of che Line of she Roor, and the
Square will both fall backward toward the middle of che Line, in the

_ fecond length.
Ff | 4. Having

peaieen

—

‘~

218 The ufe of the Line of Numberse

4. Having two extreme Numbers givens to find two mean Proportionals |
between them.

Divide the fpace between the two extreme Numbers given into -—
three equal parts, Asif che extreme Numbers given were 8 and 27,di-

vide the fpace between them into three equal parts, the feet of she
Compaffes will ftand in 12 and 18.

5. To find the Cabic Root of a Number given.

The Cubic Root is always the firft of two mean Proportionals —
between 1 and the Number given, and therefore to be found by divi= —
ding the fpace between them into three equal parts. °

So the Root of 1728 will be found tobe 12. The Root of 17280is
almoft 26: andthe Rootof 172800 1s almoft 56. |

If you fuppofe a Prick under the Number given afrerthe manner of
Arithmetical extraction, and the laft Prick co the lefe hand fhall fall
under the lat figure, asitdothin 1728, the unite will be beft placed
at 1 in the middle of the Line, andthe Root, the Square,and the Cube, —
will all fall forward toward the end of the Line. 3 7

If the la(t Prick fhall fall under the laft figurebutone, asin 17280,
the unite may be placed at 1 in the beginning of the Line, and the — |
Cube inthe fecond length, or the unite maybe placed at ro intheend
endof the Line: and the Cube in the firft lengths or if the Cube
ea of the Line, you may help your felf, as in che firft Propo-

ition. |
~ But if the laft Prick thall fall under the la(t figure but two, as 17 |
172800, then place the unite always at 10 in the end of theLine: fo —
the Root, the Square, and the Cube, will all fallbackward, andbe
found in the fecond Length between the middle and end of the ©
Line. > i |

6. To multiply one nunsber by another.

_ Extend the Compaffes from 1 to the Multiplicator 5 the fameex- |
rent applied the fame way, fhall reach from the Multiplicand roche |
Produ&. | 1 Ae |
Asif the Numbers tobe multiplied were 25 and 3.0: either ene

: the

The wfe of the Line of Numbers.” 2i9
the Compaffes frora 1 to 29, and the fame extent will give the diftance

| from 30 to 750; orextend them from 1 to 30, and che fame extent

fhall reach from 25 to 750.
| 7 To divide ont Namber by another. ,

Extend the Compaffes from the Divifor to x, the fame extent fhall
reach from the Dividend to the Quotient. |

Be if 750 were to bedivided by 25, the Quotient would be found
tobe30,

8. Three Numbers being given, tofind a fourth Proportional:

This golden Rule, the moft ufeful of all others, is performed with
like eafe. For extend the Compaffes from the firft Number to che fe-
pends che fame extent fhall give the diftance from the third to che
ourth. ;

As for example, the proportion between the Diameter and the
Circumference, is faid to be fuchas 710 22: if the Diameter be 14°
how much is the Circumference? Extend the Compaffes from 7 to
22, the fame extent fhall give the diftance from 14. to 44.: or extend
them from 7 to 14, and the fame extent fhall reach from 2 to 44.

Either of thefe ways may be cried on feveral places of this Line ;
but that place is beit, where the feet of the Compaffes may ftand

* neareft together.
9. Three Nambers being given. to find a fourth in a duplicated proportion. |

If any have daily wfe of this Propoftion, be may canfe another Line of
Nambeys to be made. |

_ This Propofition concerns queftions of proportion between Lines

_ and Superficies ; where if che denomination be of Lines, extend the
Compaffes from the firft to the fecond Number of the fame denomi-
nation: fo the fame extent being doubled, fhall give the diftance from
che third Number unto the fourth. :

The Diameter being 14, the content of the Circle is 134 sthe Dia-
meter being 28, what may the contentbe? Extend the Compaffes
from 14 to 28, the fame extent doubled fp reach from 154.to 616.

Fr 2 For

~ ; 2 x ~

Ree ER PN EP RAM ARSE EE op py OSCR en RE Si Fame ean NOT An gee rcadnag He peg reo eo) er
Be ‘ + aby ay y
7” * es y i

2200 i(is«éT ale of the Line of Numbers.

For firft, i¢ reachech from 154 unto 308-5 and turning the Compaffes i.

ence more: itreacheth from 308, unto616; and this isthe content
required. . 2°80 :

But if the firft denomination be of the fuperficial content, extend
the Compaffes unto the half of the diftance, between the firft Num-
ber and the fecond of the fame denomination: fothe fame extent (hall
give the diftance from the third to the fourth. | 3

‘The contént of a Circle being 154, the Diameter is 14.:.the con-
cent being 616, what may the Diameter be? Divide the diftance be=
tween 154, and 616 into two equal parts, then fer one foot in 14, che
other will reach to 28, the Diameter required.

10, Three Numbers being given, tofind a fourth in a triplicated proportions.

This Propofition concerneth queftions of proportion between Lines
‘and Solids; where if ehe first denomination be of Lines, extend che
_ Compaffes from the firft Number to the fecond of the fame denomina-.
rion: fo the excent being tripled, fhall give che diftance from the third
Number untothe fourth,

Suppofe the Diameter of an Iron Bullee being 4:inches, the weigher
of it was 9 /. the Diameter being 8 inches, what may the weight be2
Extend the Compafles from 4 to 8, che fame extent being tripled, —

willreach from 9 unto 72. For firft, tt reacheth from 9 unto 18; then — |

from 18 unto 36; thirdly, from36 unto 7-2., And this is cheweighe

required.

. But if the firtt Denomination fhall be of the Solid content, or ofa

the weight, extend the Compaffes toa third parrof thediftancebe-

tween the ficft Number and the fecond of the fame Denomination ; fo

the fame extent fhall give the.diftance from the. third Number unto |

she fourth,

The weight of a Cube being 72 /. the fide of it was 8 inches : the. |
weight being 9 /. what may the fide be? Divide the diftance between __

72 and 9, intothree equal parts 3then fet one footto 8, the other will, -

reach to 4, the fide required.

4

CHAP.

|
|

I
i
i
i
!

Pa >

[

= far J eee, eet isek | Mite, Oe VIA SVP ea ee Came, Ou sees el Ry peiny's
es ye &! , 4 ae oF At 4 y's 7 at) Shas
“ -. 2 > a r
‘ ‘ C

‘The ule of, the Line of Artificial Sines, 221

CHAP.) VII.
The ule off the Line of Artificial Sines.”

“W His Line of Sines hath fuch ufe in finding a fourth Proportional,

as the orciiary Canon of Sines; and the manner of finding it: is
always fuch, asin this example.

As the Sine of 90 gr. :
untothe Sine of 30 ¢r.
-. SotheSineof 20 gr, .
unco a fourth Sine. -

Extend the Compaffes from the Sine of 90 gr.unto the Sine of 30 gr.

_ the fame extent will reach from the Sine ot 20 gre unto the Sine of

Lr. 5O Mm.
Or you may extend them from the Sine of 90 gr. unto the Sine of

20 gr. the fame extent will reach from the Sine of 307. unto the

Sine of 9 gr. 50 m.and fuch is the fourth proportional fine required.
In like manner if the queition propofed were,

As the Sine of 30 ¢r. . :
unto the Sine of 52 gr. .

So the Sineof 38 gr,
co a fourth Sine...

Extend the Compaffes in the Line of Sines from 30 gr. unto §2 gr.
the fame extent fhall give the diftance from 38 gr. unto 76 gr. Or, ex-
tend them from 30 gr. unto 38 gr. the fame extent will reach from ;
52 gr.unto 76 gr. whichis the fourch proportional Sine required.

And thus may the reft of all Sinical proportions be wrought two

ways. The minutes which are wanting inthe firft degree of the Sines

may be fupplied by che Line of Numbers, as I fhew in the next.’
Chapter.

—

a2 The wfe of the Line of Artificial Tangewts:

CHAP. VIII.
The wfe of the Line of Artificial Tangents.

Be Line of Tangents hath like ufe, but commonly joyned with
the Line of Sines: the manner of working by ir, may appear by
this example: ;

As che Tangent of 38 gr. 30m.
isthe Tangent of 23 gr. 30m.
So isthe Sine of 90 gr.
co a fourth Sine.

This Propofition, and fuch others upon two Lines, may be wroughe
two ways. For extend the Compaffes from the Tangentof 38 gr.30 mm.
the Tangent of 23 gr. 30. the fame extent fhall give the diftance —
from the Sine of 90 gr. co the Sine of 33 gr.8m. Or elfe extend them
from 38¢r. 30 m. inthe Tangents unto 90 gr. inthe Line of Sines; |
the fame extent from the Tangent of 23 gr. 30m. fhall reach tothe —
Sine of 33 gr.8 m. which is the fourth proportional Sine required.

And this Crofs work in many cafes fs the better,ia regard the Tan-
gents which fhould pafs on from 40¢r.to §0 gr. and fo forward, doturn -
back at 45 gr. Thefe ewo Lines of Sines and Tangents, may ferve for
the refolution of all Spherical Triangles, according to thofe Canons
which I have fee down in the ufe of the Sector. Only two cafes the —
19 and 20 will be more eafily refolved by that which followeth inthe
laft Chapter of this book. me

Or if at any time one meet with a Secant, Let him account the Sine
of 8ogr. foraSecantof to gr. and the Sine of 70 gr. for a Secant of
— 20gr. and fotake the Sine of the Complement inftead of the Secant,

Asif the Propofition were,
As the Radius
~ cotheSecant of srgr- 30m,
So the Sine of 23 gr«Z0 m.
toa fourth Sine. -

Extend

rae ape

__ Extend the Compaffes from the Radius that is the Sine of 90 gr. to
| the Sine of 38 gr. 30m. the fame extenc will give the diftance from
the Sine of 23 gr. 30 m. bothto the Sine of 14 gr. 22 mrothe Sine of
39gr. 50m. Butinchis cafe, che Sine of 39¢r. som. is the fourth
required. For the firft number being lefs chan the fecond, chat is, the
Radius lefs than the Secant,the Sine of 23 gr. 30 m, which is thethird,
muft alfo be lefs than the fourth. |
If the fourth proportional number fhall at any time fall out of the
Line, by reafon of the minutes that are wanting in the firft degree, it
may be fupplied by refolving the third Number given into minutes,
and then working by the Line ot. Numbers.

As if the Propofition were,

AstheSine of 90 gr. | K
tothe Sine of 10 gr. salt,
So the Sineof 5 gr.
_ toa fourth Sine.
- Orthe Tangent of § gr. ) : f
toa fourth Tangenr. |

Extend the Compafies from the Sine of 90 gr. untothe Sine af
_ Togr, the fameextent will reach from the Sine or Tangent of § gr.
| -beyond the end of the Staff. Wherefore I refolve thefe 5 gr. into
300m. and find the former extent toreach in the Line of Numbers
from 300 m, unto $2 m, and fuch is the fourth proportional re-
quired.
If the extent from the Sine of 90 gr, unto the Sine of 10 gr. be too
large for the Compaffes, we may ufe the Sine of 5 gv. 44.. inftead of
the Sineof 90 gr. ? |
__ And ‘fo extending the Compaffes from the Sine of 5 gr. 44 m. unto
the Sine of 1© gr. we fhall find the fame extent to reach in che Line of
Numbers from 300 unto 52 as before, |
And by the fame reafon we may ufe the Tangent of 597. 43 m. in-
flead of the Tangent of 45.g7. as_1 further fhew in the nexs Chapter. _

CH AP:

| The afe ofthe Line of Artificial Tangents; = 203

a, oes ne ren See ee Pe ee ag
© The ufe of the Line of Sines and Tangents;
‘CHAP. IX. a
The fe of the Line of Sines and Tangents joyned with the Line ‘ |
: oe of Numbers, ee

my Fi Lines-of Sines and Tangents have another like ufe joyned with |
the Line of Numbers, efpecially inthe refolucion of right Lined,
Triangles, where the Angles are meafured by degrees and minutes,and
the fides meafured by abfolute Numbers,whereof, I will fet down thefe a
-Propofitions.

1. Having three Angles and one fide, to find the two other fides.

ee = ee eee

Tf itbea Reétangle Triangle, wherein one fide about the right An= ~—
ye: ‘gle being known, it were required to find the other. This may be
found by. the Line of Tangents, and Line of Numbers. For,

As the Tangent of 45 gr. | |

_ _ To the Tangent of the Angle oppofite to the fide required ; a
So the Number belonging to the fide given, a
Tothe Number belonging to che fide required. 2 ae |

| res As in the Rectangle —
pe ef 3/3 A BC, knowing the An- a
5+—__Ip gle CAB to be ogr. |

| 15 m. and the fide A Bro © |

be 135 parts, if it were required to find the other fide BC abourthe a)
right Angle, | z|
Extend the Compaffes from the Tangent of 45 ¢r. untothe Tangent
of 9 gr. 15 m, the fame extent will reachin theLine of Numbers,
from 135 unto 22, and fuch isthe length of thefide BC. Or inthe
crofs work, extend the Compaffes from the Tangent of45gr.unth 135
inthe Line of Numbers, the fame extent will reach from the Tangent
of 9 gr.1§ m. unto 22 inthe Line of Numbers. sii
If this extent from the Tangent of 45 gr. tO g gr. 15m. OF, 135 |
parts, 0¢ too large for the Compaffes, you may ufe the Tangent of § gre

43 %.

|

oo «(|

. _ joyned with the Line of Numbers, = 2.25

43 m. inftead of che Tangent of 45 gr. becaufe both alike anfwer to

10, @&c. parts inthe Line of Numbers.

_ — And then either extend the Cempafs from 5 gr. 43 m. unto 9 gr.

| 1§ m in the Line of Tangents, the fame extent will reach from 135

_unto 22 inthe Line of Numbers: or elfe extend them from the Tan-

| gencoft § gr. 43 m. unto 135:inthe Line of Numbers, the fame extent

will reach from che Tangent of 9 gr.15 m.Unto 22 in the Line of Num-

| bers, as before. . . |

_ Ihnlike manner, in the fame ReCtangle A BC, knowing the Angle
AC Beobe 80gr. 45 m. andthe fide BC to be22 parts, it were re-

| quired to find the other fide BA. You may ufe the Tangent of 84¢r.

17m. inltead of. the Tangent of 45 gr. and fothe fide B A willbe

_ found tobe 125 parts. eee ee

_ Thisholdeth for finding of the fides of Rectangle Triangles, but

generally in all Triangles, whether they be right or obtufe Angles, ha-

ving three Angles and one fide, we may findthe two other fides by the

| Line of Sinesand Lineof Numbers. a :

_ Asthe Sine of an Angle oppofiteco the fide given,

- istoche Number belonging to chat fide given 5

So the Sine of the Angle oppofiteto the fide required,

“to the Number belonging to the fide required, .

\

-_ As in the example of the fourth Chapter of this Book, where
knowing che diftance between two {tations at Aand D to be 100 paces, .
_ the Angle B A C to be 43 gr. 20m, andthe Angle BD C co be 58 gr.
itwas required to find the diftance AB. feu
Firft, having thefe two Angles, I may find the third Angle A B D to
be 14 ¢7.4.0 m. either by fubftraGion or by Complement unto 1280.Then
_ inthe Triangle BA D, [have three Angles, and one fide, whereby I
| May find both A Band DB.

_ Iksow the Angle ABD oppofite tothe meafured fide A D, to be -
_ 14 gr. 40 m, and the Angle A D Boppofite to the fide required, to be
_ 122gr. wherefore I extend the Compeffles inthe Lineof Sines, from
| 14 gr, 40 m. unto 122 gr.or (which is all one) to 58gr. ( for after
_ 90¢gr. the Sine of 80 gr.isalfo che Sine of 100 gr, and the Sine of 70 gre
_ the Sine of r10 gr. and fo inthereft ) fo fhall I find the fame extent co
- reachin the Line of Numbers, from 100 unto-335. And fuch is the di-
_ ftance required between Aand B, |
| Gg In

ee CE hc CTA ENE Pee ye any ee
ah * BY thin Rinne Sy

6 The ufe of the line of Sines and Tangents; =
. ~ Jn like manner I extend my Compaffes from theSine of 14¢r.

FORE mh i ,
RY ee ’

Pe 4 Ae

od
- *

<5
40m. tothe Sine 43 gr. 20 a. the fame extent will reach inthe Line — |
ie of Numbers from tooto271. And fuch is the diftance between D
a and B. i ° bets ; Bie
, E = 7 » |
sn : ‘ yy qi 2 - ; ee
° B Wa e
y i f ~Q wit
fom iW, AY i 4
NV, fy i, : a.
LSS ft
ry ¢ 2%, ¥
$ foo 7.
: - : 4 a
C) s = \ y :
$ : %
wy : ‘vy : 2 s &
: fo if $ qi %
| AL ( Pe is : * :
A ae Ne ; : a
% © ms o ND
Ay 2 », J 5B | 3 : v
100 D 2A. ; a,
+ a ei Eg anes Wi
@r incrofs work, I may extend the Compaflesfrom14 gr. 40m,

inthe Sines, unto 100 parts in the Line of Numbers, fo the fame ex- _

tent will give the diftance from 58 gr. to 335 parts, and from 43 gr
2.0 m-. tO 271 parts.

OW §
ta
¥

2. Having two fides given, and one Angle oppofte to either of thefe fides |
to find the other two Angles and the third fide, a

As the fide oppofire to the Angle given, | as

is to the Sine of the Angle given: |

So the other fide given,

to the Sine of that Angle to which itisoppofite, ae
So inthe former Triangle, having the two fides, A B 335 paces,and_
A D too paces, and knowing the Angle AD B, which is oppofite to
the fide A B,to122gr. I may find the Angle AB D, which is oppofite _
to.

ed

-. Goyned with the Line of Numbersi = 229

tothe other fide AD. For if Textend the Compaffes from 335 unto

» roo inthe Line of Numbers, I fhall find the fame extent to reach in
_ the Line of Sines from 1 22 gr.to 14 gr. 40 meand therefore {uch isthe
Angle ABD. att , 7

_ Thenknowing thefetwo Angles ABD and ADB, I may find the

third Angle B A D either by fubtraftion or by Complement to 180, to

be 43 gr. 20m, and having three Angles and two fides, I may well
-find che third fide D B, by the former Propofition.

This may bedone more readily by crofs work. For if I extend the —
_ Compafies from 335 parts, inthe Line of Numbers, to the Sine of

122 gr. the fame extent will reach from 100 parts to the Sine of 14 gr.

46 m. and back from 43 gr. 20m. to 271 parts; and fuch is the third
fide DE.

_ +3. Having two fides and the Angle between thems, to find the two other

Angles and the third fides

If the Angle contained between the two fides bea right Angle, the —

other two Angles will be found readily by this Canon.

_ _ Asche greater fide given,

isto the leffer fide:
Sothe Tangent of 45 ¢r.
to the Tangent of the lefler Angle.

Sointhe Re€tangle triangle, AI B, knowing the fide A I to be 244,

and the fide 1B 230: if I extend the Compaffes from 244 to 230 in
the Line of Numbers, the fame extent willreach from 45 gr. to about
(43 gr. 20m, in the Line of Tangents; and fuch is the leffer Angle

BAI, and che Complement 46 gr. 40m. fhews the greater Angie

ABI. The Angles being known, the third fide A B may be found by
_ the firft Propofition. | ; |

So likewife inthe example of the third Chapter of this Book, con-
cerning taking of Angles by the Line of Inches, where the parts in-
tercepted on che Staff being 20 Inches, and the parts on the Crofs 9
Inches, it was required to find the Angle of the Altitude, For,

I may extend the Compaffes inthe Lineot Numbers, fron 20 un-

tog, the fame extent will reach in the Line of Tangents from 45 gr.to
-24gr.14m,

Gg2 Or

POA Ne ee fee aL UT LA Foe By RO EG EE OMEN OR NIST CNL eM PO 2 us Pep aed Se AY
i a Se A R wt FTA SANE A Le K Vow 7 voice ts ha XS
by ¢ RF ee ss ae a 3 : * Me te . - i

| 228 —- Theale of the Line of Sines and T: ARgENE Sy

NS ESE ss Ne SIN Pht SOO ROL. 2 ea dy RRR FOR Sa ea

Or in crofs-work,

I may extend the Compaffes from 20 parts in the Line of Numbers, _

tothe Tangent of 45 gr. the fame extent fhall give che diftance from —

9 parts, unto the Tangentof 24¢r. 14m, mm
And fuchis the Angle of the Altitude required.

If che parts interceped on the Staff being 20 Inches, and the parts —
onthe Crofs 9 tenth parts of an inch,it were requiredto find the An. —
gle of che Altitude. Here the Anglewould be much lefs, and theg ©

would fall out of the Line of Numbers..

To fupply this defe&t, Iufethe Tangent of sgr. 43 m. inttead of i
the Tangent of 45 gr. And chenif I extend the Compaffes inthe Line —
of Numbers from 2ounto9, the fame extent willreach inthe Lineog —

Tangents from § gr. 43 m, unto 2 gr. 35.

Or in Crofswork, if I extend them from 20 parts inthe one Line _
of Numbers, unto the Tangent of 5 gr. 43 m. the fame extene will —
give the diftanee from g inthe Line of Numbers, unto the Tangent of —

2 gr. 35 Mm. :
And fuch is this Angle of the Altitude required:

But if it be an oblique Angle that is contained between thetwo fides — |
given, the Triangle may bereduced into two Reétangle Triangles, ‘|

a

and then refolved as before.

Asin the Triangle ADB, where the fide AB is 335, andthefide _

_ A Dioo, and the Angle BAD 43 gr. 20m, If Iles down thePer-
pendicular, DH upon the fide AB, I fhall have two Re&angle Tri-
angles, AH D, DHB;; and in the Reétangle AH D, the Angle ae
A being 43 gr. 20m. the other Angle AD H willbe 467. gom,
and with thefe Angles andthe fide A D, I may find both AK andDH,

by the firft Propofition.

Thentaking A H out of AB, thereremains HB for the fide ofthe

Reangle D H B, and therefore with this fide H Band the other fide

ns H D, I may find both the Angle at B, and the third fide D B, asinthe

former part of this Propofition.

Or I may find the Angles required, without fetting down any Per-.

pendicular,. For,

As the fumof the fides;
istoche difference of the fides :
So the Tangeat of the half {um of the oppofite Angles,
co the Tangent of half the difference between thofe Angles.

| ‘More concerning Chords,.

a 1 Cee ee 8 Ee Ss PRD? oye.
To" AeA” lle a4 Ral cesite bay: i Rats Uys Rae tet a aale eee Kener
ME ARN y SOTA REO TM AMC EGE NTT Coreg Ronse woe y OBR
Natit Oi eae: LD Ry NY we ae)

Bye eyed aks Ye et fF ct | é vu

~ s 4 ,: ned

| __Asinthe former Triangle AD B, the famof the fides AB, A D, is

| 435, and the difference between them 235 ; the Angle contained
43 gr, 20 m. and therefore the fum of the two oppofire Angles

|. 136gr.40m, and the half fum 68 gr. 20m. Hereupon I extend the

and fuch isthe half difference between the oppofite Angles atc Band

D, This half difference being sdded to the half fum, doth give 122 gr.

for the greater Angle AD B: and being fubtracted, it leaveth 14 gr.
40 m. for the lefler Angle A B D,then the three Angles being known,
_ the third fide B.D may be found by the firft Propofition.

| ; ! |
4+, «Having the three fides of a right Line Triangle, to find the three
| eAngles.

| Let one of the three fides given be the Bafe, but rather the sreater.
fide, thatthe Perpendicular may fall wichin the Triangle ; thea ga-

ther the'fum, and difference of the two other fides, and the propor-
tion will holds. |
|

As the Bafe of the Triangle,
is to che {um of the fides :-
| So the difference of the fides
ae. to a fourth, which being taken forth of the Bafe, the Perpen-:
dicular fhall fall on che middle of the remainder.

_ As inthe former Triangle A D B, where the Bafe A B is 335; the
fum of the fides AD and D B 371, and the difference of chem 171,

| If I extend the Compaffes in the Line of Numbers from 335 unto 371,.
I fhall find the fame extent to reach from 171 unto 189. 4, This fourth
_ Number I take out of the Bafe 335. 0. and the remainder is 145, 6,

the half whereof is 72. 8, and doth fhew thedi(tance from A unto H,

‘where the Perpendicular fhall fall, from the Angle D, upon the Bafe
_ AB, dividing the former Triangle A D B into two right Angle Tri-.
| angles, D-H’A’and’ DH B, inwhich the Angles may be found by the
| fecond Propofition. | | |

____ And this may fuffice for the right Line Triangles, but for the more’
| eafie protraction of thefe Triangles, Iwill fer down one Propofitiom

5, Having:

5 jaymed with the Line of Numbers. ang

| Compaffes in the Line of Numbers from 4350235, andI find them. .
., toreach in the Line of Tangents from 68 gr. 20m, Unto 53 gr,40m,.

aes rat i : Bs 7 Away ai a thas Ni | ‘s i nS h het W Brin tos Dye bi |
yes $8: BG es Re Mer Aa Lie ia el oer ne aE Aine wt
230. The ufe of the Line of Sines and Tangents, &c,

ea eae : , i ' ) oy wil tare ‘ rye ey a

5. Having the Scemidiameter of a Circle, to find the Chords of tony 4

Asche Sine of the Semiradinsof 32 gre i : &
| ro the Sine of half the Ark propofed : _ teh :
aes So is the Semidiameter of the Circle given,
5 tothe Chord ot the fame Ark.

As if inthe protracting the former Triangle A D B, ie were re-
quired to find the length of a Chord of 43 gr. 20 m. agreeing
to che Semidiameter A E, which is known to be three Inches.

nie _ Thehalf of 43 gr. 20 m. is 21 gr. 40 mm. wherefore I extend the
_.___. Compaffes from the Sine of 30 gr. to the Sine of 21 gr, 40 m.
- and I find che fame extent to resch in che Line of Numbersfrom _
e. 3. 000 parts to 2.215. which fhews, that the Semidiameter being .
{ oa Inches,the Chord of 43 gr.20 m. willbe 2 Inches. and 215 parts — )
Resin) of 100, *
roe Inlike manner the Chord of 58 gr. agreeing to the fame Semidia- _
meter, would be found to be 2 inches and 9e9 parts. For the halfof
ae 38 being 29; if I extend the Compaffes in the Line of Sines from 30
Oe gr.to 29 er. the fame extent will reach: inthe Line of Numbersfrom _
mt % 3. 000. unto 2. 909. pS
ae Or inCrofs work, if lextend the Compaffes from the Sineof 30. —
a gr. to 3.000 in the Line of Numbers, I fhall find che fame extentto ~
cae reachtrom 21 gr: 40m. tO 2, 215 parts, and from 29 gr. t02.909
ie parts, and from 7 gr. 20 m, to79§ parts ; for the Chord of 14 gr.
ss 40m, forthe rhirdAngleABD. = a

1.) es |

CHAP. X,. |
‘The #f¢ of the Line of verfed Sines. |

i io His Line of verfed Sines isno neceffary Line. For all Triangles,
1% Ton right lined and fpherical may be refolved by the three for-
mer Lines of Numbers, Sinesand Tangents; yet I thought good to

put it on the Staff-for the more ealie finding of an Angle having
_ three fides, or a fide having three Angles of a fpherical Triangle

.

| given.

~ ‘Suppofe the three fidesto be, one of them 100 gr. che other 78 gr.
and the third 38 gr. 30. and lec it be required to find the Angle,
whofe Bafe is 110 gr. | my
I firft add them together, and from half thefum fubtract the Bafe,
noting the difference after this manner. EE

The Bafe 110 gr. Om.
The one fide 46 ON
The other fide 38 30

The fum of all three 226. 30

The half fum Cc 03) PhS

The difference Se LS

For fo the proportion will hold.

tAstheRadius —_

to the Sine of one of ehe fides:
Sothe Sine of the other fide,

toa fourth Sine.

2 As this fourth Sine, | ;
| to the Sine of the halffum :
eee Soxhe Sine of the difference
| ~ ** cog feventh Sine.

wee

NE. Pres
pie,
Bey te
Je

Oe he ae a ANT ae od ©
vate eA. NWA eh ae ae oo ame \
NS item EVES epee
Dae

Sha ee

“The nfe of the Lineof verfedSines,

3 The mean proportional between this feventh Sine and the Radius, _
will fhew the Sine of the Complement of halfthe Angle required. © ~
. Thisdoue, I come to the Scaff, and extend the Compaftes fromthe _
Sine of go gr. to the Sine of "78° gr. whiclris one of the fides ; andap-
plying this extent from the Sine of the other fide 38 gr. 30m. I find
it coreachtoa fourthSine, about 37 gre30 wm. From-this fourth Sine
of 39 gr. 30 m. Lextend the Compafles again, to the Sine of thehalf
fum 113 gr. 15 m. ( which is all one wich the Sine of 66 gr. 45 m. )
and this fecond extent will reach from the Sine of che difference 3 gre

15 metothe Sine of 4 er. 5.4 mm, . !

Thento find the mean proportional Sine between this feventh Sine -

3 hs. ¥ Sa Seve
ree i i} VRS ee?” >
‘ a Fea eh Rea
: 1 YA Ny
" i
.

of 4¢r.54.m.and the Sine of go gr. I might divide the {pace between

them into two equal parts, and fo] fhould find the Compaffes to ftay
at 17 gr. whofe Complement is 73 gr. and the double of 73 gr. is146
grethe Angle oppolite to 110 gr. which was required. i:
Bue becaufe this divifion is fomewhattroublefome I have therefore
added this Line of Verfed Sines, chat having found che feventhSine
you might look over againftit, and there findthe Angle. And foin
this example having found the feventh Sine to be 4.gr. 54m. over
againft this Sine you fhall find 246 gr. in the Line of Verfed Sines for
the Angle required as before. 4

_ >.
¥ .S

THE - at
SECOND BOOK “
tka: Sy OF THE c
CROSSSTAFF.:

as:

cD da bhacsd Slob slthans a holedadat tT tabe tay COT ery

ie Of the ufe of the former Lines of Pro-'
« pofition more particularly exemplified
in feveral kinds. .

2|

fai He former Book containing the gene-
WN ral ufe of each Line of proportion,|
PQS may be fuficient for all chofe which

“1

x
SS SS
SY NER know the Rele of Three, and chelm
doctrine of Triangles.

But for others, I fuppofe ic would be more dif
Geuleto find either the Declinationof the Sun, o:
his Ampl cude, or the like, by chat which harh|
| been faid in the ufe of the Line of Sines, unlef-} |
they may havethe particular proportions,by which} .
fuch propolitions are to be wroughe.

And therefore for their fakes I have acjoyne: |
this fecond Book, containing feveral proportion:
for propolitions of ordinary nfe, and fee them |
down in fuch order, that the Reader conlidering ,
which is the firft of the three Numbers g ven, may fx {Ils
_eafily apply them to the Sector, and alfa refoly: : x

- themby Arithmetick, beginning with thofe whicl

require help only of the Line of Numbers. So i

aN:
Hh CHAP.

p a

®

ko

6 aN Aw »®

234

CHAP, I. |

The nfe of the Line of Numbers in broad mealwre, [uch as Boards
: | Glafs, and the like. |

J ‘He ordinary meafure for breadth and length are feet and inchess.
each foot divided into §2 inches, and every inch into halves and

quarters,which being parts of feveral denominations doth breed much
trouble both in Arithmetick and the ufeof inftruments.

-Forthe avoiding whereof, where I may prevail I give this counfel,
that fuch asare delighted in meafure would ufe feveral Lines, firfta
Line of inch meafure,, wherein every inch may be divided into 10.
or 100 parts 3 fecondly, a Line of foot mealure, wherein every
foot may be divided into 100 or 1000 parts, both which Lines may
be fet on the fame fide of a two foor Ruler, after this or the like.
~ manner. ’

Then if chey.be to give the content of any Superficies cr Solid im
inches, they may meafure che fides of it by the Line of inches and
parts of inches; bucif chey be to give the content in feet, it would

be more eafie for them to meafure thofe lides. by.the foor Line and his.

arts.
a Bor example, let the length of a- Plain. be 30 inches, and the.

breadth 21 inches and ©; of aninchs this lengrh multiplied into the
breadth, would give the content to be 648 inches: but if Iwere —
to find the content of the fame Plane in teer, Iwonld meafure che
fides of it by the foot Line and his parts; fo the length ‘would prove “—
fo be two feer$2,, and the breadth one foot ,*°., and the length
multiplied by the breadth, curting off the four. lait figures, for the.
four figures of the parts, would give the content to be 4. sco)
| | whic

The ufe of the Line of Numbers in broad mesfure, 235
_ which is 4 foos and 5000 parts of a foot, divided into 10000
parts. ; i'e

| 21. 6 2. 50
[ 30. © 1, 80
948,00 20000
250
4 5000

f The like reafon holdeth for Yards and Ells, and all other meafures
divided into 10, 100 or 1000 parce, ae
This being prefuppofed, che work will be more eafie both by

Arithmetick and the line of Numbers, as may appear by chefe Pro-
| pofations. |

S EC Te I.

l) Of bbe Manfuration of Oblong Superfeles, and Tréangles.
| a
1. Having the breadth and length of an Oblong Superficies given in inch-

~ meafure, to find the content in inches. .

| A S one inch unto the breadth in inches,
| FA Sothe length in inches unto the content in inches.

“i cs

oe
asoe”
eee?
oo!
{wees
awe
ig 00?
PT tbe
use?
ase
au8®

__ Suppofe in che Plane AD, the breadth A C to be 30 inches, and the
Teugeh A B tobe 183 inches 5 extend the Compaffes from 1 unto 30,
‘the fame extent will reach from 183 unto §49¢ ; or extend them from
~ Eunto 183 the fame extent will reach from 30 unto 5490. Sobotk

Ways the content required is found to be $490 inches,
|

As I unto 30; foare 183 unto $490. ‘
Hh 2 2, Having

,) tae

236 8 The afeof the Line of Numbers.

“a. Having the breadth and length of any Obleng Saperficies given in
inches, to find the content in feet. Soe a yau

As144.inches untothe breadchin inches :
So the length in inches unto che content in feer. taal, a
And thus io the former Plane A D, working as before, the content —
will be found to be 38. 225, which is 38 foot and 4 of a foor.
As 144.untd 30, fo are 183 unto 38. 125,

3. Having the length and breadth of any Oblong Superficies given in foot
wsealure, to find the content in feet. -

As 1 foot unto the breadth in foot meafure :
Se the length in feet unto the content in feet, :
And thus inthe former Plane A D, che breadth will be ewo feet 50°

parts, and the length 15 feet 25 parts; then working as before, the
content will be found to be 38. 125. |

As Lunto 2. 50: fo are 15, 25 unto 38. 125,.

4. Having the breadth of any Oblong Superficies given in inches, and the |
length in fout meafure, to find the content in feet.

As 12 Inches tothe breadth in inches:
Sothe lergeh in feee co the content in feer,.
So allo in the former Plane, the content will be found to be 38.
I25- .
As the 12.unto 30: foare 15.25 unto 38-1 25%

-
ps

j. Having the breadth of an Oblong Superfcies given in inches. to ids
the length of a foot fuperficial in inch mealure, } i 1

Asthe breadthin inches, unto 144 inches :
So 1 foot unto the length in inch meafure,

So the breadth being 30 inches, the length of afoot will be found: —
to be 4 inches 80 parts, the length of two feet 9 inches Soparts.
As 30 unto 144; {0 are runto 480.. | pelt |

6. Having

aa }

-

Oda bysad meafares 8

| 6. Having the breadth of an Oblong Supesficies ginen in feels to find the —
, length of a foot fuperficlal in foot meafure. % ia hs

As the breadth in foot meafure to 1 foot:
So the number of feet tothe length in foot meafure. ,
So the breadth being 2 feet 50 parts, the length of a foot will be
found to be 40 parts, the length of 2 feet 80 parts, and the length of -

3 feet 1 foot 20 partssa’c.
As2sounto 1: foare I unto @, 40,

9. A four faded Superfickes having any of the two fides Parallels, to find
the Area. 7

Add the two Parallel fides together, and take the half,then fay,

5 1 |
is to the half fum of the two Parallel fides :

So is the breadth (or length ) |
cothe Area,

| Soin the four fided figure A having -one of the paraffel fides in-.
Jeneth 23. 25 foor, and the other, ro. 7§ foot, the fum. of them is
| 43.00 foor, ( the half whereof is 21. §0foot ) andthe breadth 14. ¢,
| fhe Area or content of this Superficies will be found to be 311,75.
\-foor. - , .

© Exrend che Compafles from 1 to21.50 (the mean length ) the
- fame extenc will reach from 14.5 (the breadth ) to 321. 15 the Area.

or content.

"BL Tafind the Areaer content of a Triangle, the longe/t fide and the Per-.
pendicular being given. eR |

| Asi,

_ > istothe half length of the Bafe :.

| _ So isthe length of she Perpendicular.
| to the Content or Area. rth

Fe ee eae By MR Pe he tae I PLS, tee Or ee ee
Bo ie Sit | AE oy Sack ale wis able: +H
as : Ly

re; Hag! The ule of the Line of Numbers te ee
; So the Triangle B, having she Bafe 32

foor, and the Perpendicular 2¢ foot, the
Area will be found to be 400 foot. ©

Or, extend the Compaffes trom 1 to 12, 5.
ony Yea (half the length of the Perpendicular ) the
fame extent will reach from 32 ¢ the whole
Bale) to 400 as before.
Or, extend the Compaffes from 1 to 32, the fame extent wil! reach
from.25 to800,the double Area. :

9. The fide of an Equilateral Triangle being given, to find the Arid

As rooo,
15 (0 4.33.01 , |
So is the Square of che fide of the Triangle,
tothe Area. |

So the fide of an Equilateral Triangle being 17. 5 foot, the Area ~

will be foumid co be 132. 61 foor.
Excend the Compaffes from 1e00 to 433. ot, the fame Extent will

reach from 306.25 (the Square of the {ide of the Triangle) to 132.
61, the Area. |

10. 79 find the Area of afour fided figure, whofe fides are either egal
nor paralel one to the other, Which figures are called Trapexias.

All four fided figures whofe fides are neither equal nor parallel,

muit be reduced into two Triangles, by drawing a Diagonal Line

from any one Angle to its oppofite; upon which Diagonal twoPerpen-

diculars mutt be let fall ; chen,

AsI,

isco half the length of the Diagonal :
So isthe length of both the Perpendicular,
To the Area, or Content.

Se

;

: _ Extend the Compaffes from to.16, (half
D5: the Bafe ) che fame extent will reach from
B i 3§ (the Perpendicular) to 4oothe Area.

, lag | in broad meafures 239

___ So inthe Trapezia C, the

_ Diagonal is 68, ( the half of

| 1t1s 34) and che tivo Per-

| pendiculars are 32 and 19,

their Sumisg1. Tnen,

_ Extend tne Compafies

| from gto 34. ( half che Di-
-agonal ) che fame extent

will reach from §1 (che fum
of the Perpendicuters ) co

| 1734.the Area, ieee

Or, extend the Compaf-
fesfrom 1 to 63 ( the Dagonal ) the fame extent will reech from 25. 5
_Chalfche fum of the two Perpendicalars ) tor7 24. asbefore.

In all other right lined iigures, of how many fides, or how Irie-
“gular foever, before they can be meafured they muft (by drawing of
Lines fron Angle to Angle ) be reduced into Triangles or Trapezias,
and fo be meafured by thefe ewo laft Precepts.

And here note, Thag when any irregular figure is thus reduced
into Triangles, the number of Triangles will be lefs by ewo
than che nuraber of the fides of the irregular figure,“

1; Having the length and breadth of . an Oblong Superficies, to find the
fide of 4a Square equal to the Oblong. | ee f

Divide the {pace between the length and the breadth into two equal
‘parts, and che foot of the Compafles will ftay at the fide of the

{quare. | +
So che length being 183 inches, andthe breadch 30 inches, the fide

ofthe fquare will be found to be 74 inches, and almoft 10 pares of

100,, |
Or the breadth beimg 2 foot and s0 parts, the length 15 foot and

25 parts, the fide of che fquare will be found. to be about 6 feet and
17-parts. ,

As 30 unto 74,10 : fo are 74, 10 unto 183, 027,.
And as 2550. unto 6,174; foare6, 174 unto 15, 247.

SECT,

“}

Ts aay, TSR 5 BRO TE Aad Ya ae r 2 i=
we TG i eran SAA Ome oe tee mr 4

: 246 rhe ufe of the Line of Nansbers

SECT. Te

Of the Menfuration of Regular Polygons.

Y Regular Polygons are meant all fuch figures whofe fides-and
Bacets are above four, and are allequal. Asche i
| Pentagon Yoo a?

‘Hexagon | 6

Hepragon | which confi-g 7 Equal fides and
O@agon 7 fteth of 8 Angles.
Nonagon AHOe' ss

Decagon .J WLIoJ

And the Area of any of thefe Regular Polygons is equal toa Paral-
lelogram, whofe length (hall be equal to half tue Perimeter,and whofe”
breadth equal toa Perpendicular drawn from che Center of the figure

-cothe middle of any of the fides of the Polygon.

on, The fide and Perpendicular-of 4 Pentagon being given, to find the

~ Area.
Ast ;
is to the Perpendicular
(8,258.
So is half the Perimerer
(3010
To the Area 247. 74

inches.

Soin this Perttagon, where
_ thelideC D (and fo all che
reft_) contains 12 inches, and:
the Perpendicular H Z,8,258
inches,che Area will be found

| tobe 247. 74.
Extend the Compaffes from 1, to 8. 258, (the Perpendicular ) the
fame extent willreech from 30 (which is half the length of all ches
fides ) to 247. 74 the Area. 2. The

be dal

given, to find the Area.

ime Ast Vr $ ¢
isto the Perpendicular 14. 48,

| So is the Semi-Perimerer 48.

(0 695. o4 the Area.

| So a regular Polygon of 8 fides, each fide. containing.12 inches

| andthe Perpendicular 14 48 inches, the Area thereof will be found

| to be 695. 04. Pe |

Extend the Compaffes from t to 14. 48 the Perpendicular, the fame
extent will reach from 48 (half che Perimerer ) to 695 04. the Area.

_ And inthis nature, may any Regular Polygon, of what number of
{ides foever, be meafured. 3

Secr. Hl.
Of the Menfaration of Circles:

“W ' He Proportion of the Diameter of a Circle to its Circumference
HL as 7 is to 22. but Ladolph Van Culea comes fomewhat nearer,
allowing theDiameter tothe Circumference to be(near Jas 113 to 355:
which proportions I fhall ufe in the following Problems. .

| 1. The Diameter of a Circle being given, to find the Circumference.
|. .
Ast13 isto355:

So is the Diameter to the Circumference.

- So the Diameter of a Circle being t5 inches, the Circumference
|

will befoundcobe 47.12, .
|. Extend the Compaffes from 113.

to 355. the fame extent will reach
frota 15 the Diamerer, to 47, r2inchesthe Circumference.
| $ 4

li 2.7be

i.

inboard meafures 241

2. The Side and Perpendicular of an O€8agon ( or figure of 8 fides) being

ee

242 Tbe ufe of the Line of Numbers
2. The Circumference of a Circle being given, to find the Diameter.

AS 355 1 tO 1338
So isthe Circumference to the Diameter.

So the Circumference being 47. 12. the Diameter will be found to
be 15 inches. |

Extend the Compaffes from 355 to 113, the fame excent will reach |
backwards from 47.12 to15.

3. The Diameter of aCircle being given, to find the Area

As 28 | J
isto 22: |

So is the fquare of the Diameter 225
torhe Area 176, 6F.

So the Diameter being 15 inches, the Area will be.tound to be
176. 61. | |

Extend the Compaffes from 28 backwards to 22, the fame extent
applied ( the fame way ) will reach from 225 ( the Square of the Dia- —
meter ) to 176. 61, the Area, :

4. The Area of a Circle being given, to find the Diameter.

~ As 22
isto 28: ,
So isthe Area176,61,
ro tLe Square of che Diameter 225.

So the Area of a Circle being 176. 61 inches, the Diameter will be
found to be 15 inches. del

Extend the Compaffes from 22 to 28, the fame extent will reach
from 176. 61 to 2247. 08 the Square of the Diameter, the middle
way upon the Line between 2247. o8 and 1, is 15 the Diameter.

5. The.

in board meafuves 243
5. The Circumference of a Circle being given, to find the brtar

| As88

isto 72 :

So is the Square of the Circumference 2220. 29
cothe Area 176. 61.

So the Circumference of a Circle being 47. 12, the Area will be |

found to be 176. 61 inches.
Extend the Compaffes from 88 to 7,the fame extent will reach from »

2220, 29 ( the {quare of the Circumference ) to 176. 61 the Area.

6.The Areaof aCircle being given, to find the Circumsference.

As7
isto 88 :
So isthe Area 176. 61.
tothe Square of the Circumference 2220, 29.

Sothe Area of a Circle being 176.61, the Circumference willbe

found to be 47. 12.
- Extend the Compaffes from 7 to 88, the fame extent will reach from

176. 61, the Area to 2220. 29, the Square of the Circumference,
the half diftance between 1 and 2220, 29, is 47. 12 the Circum-

ference.

9. Having the Diameter of a Circley to find the fide of a Square equal to
that Circle.

As 10000 to the Diameter :
* ~ $0 8862. unto the fide of the Square.

So the Diameter of aCircle being 15 inches, the fide of che {quare

| will be found about 13 inchesand 29 parts.
| As 10000 unto 8862: fo are 15 unto, 29.

Tt2 ; 3, Havisg

244. The ufe.of the Line of Numbers >

oe i

8. Having the Circumference of a Circle, to find the fide of a Square
equal to the fame Circle. i ‘
As10000 '0 the Circumference: |

So 2821 to the fide of the Square.

Sothe Circumference of a Circle being 47 inches
of the Square will be about 13 inches 29 parts. .
As 10000 unto 2821: foare 47, 13 WNtO 13,29.

13 parts, the fide

SECT, TY.
Of the Menfuration of Land by Perch and Acres...

t, Having the breadth and length of an Oblong Superficies, given in
Perches, to find the content in Perches,
§ 1 Perch, to the breadth in Perches:
So the length in Perches, to the content in Perches.

So in the former Plane A D, if the breadech A C be 30 Perches,
and the length A B 183 Perches,

Perches.

2, Having the length and breadth of an Oblong Superficies given is Pere
ches, to find the content in Acres,

As 160, tothebreadchinPerches: | | ‘
So the length in Perches, to the con-ent in Acres,

So in the forver Plane AD, the content willbe found to.be 34
- Acres, and 31 Centefmes, or parts of 100,

As 160, "nto 30: So are 183, UNIO 34,

|
4

r
f;

To augment a Stwyerficles in a proportion, Re
Lo dimsinifh a Suprficies in a proportion givens

ance ee
. Inch.
—— Birt
Inch. I
Cent. 62,7264
Foor. 144
Pace. 3600
|Perch. 39204
———— eee Re
Chain, 627764
Acre. 6272640
| Mile. 4014489600
SQuaic.

A Table for the ufe of the Chain.

TS I Pr apr mate Pfau seen

i

PEE Cees: EE) <span, SE

x

Cenc.

Ce@lererere)
ey TS

64000000

———— eee

een

27878400 |

10, 89

17424

174254

TIT 5136

| 7920 |—
| c3_ | ose |e
pea eee] Sas
| 232 P32} 1050
Ere
Sen ae

| I 10 | So
2 is are
| ——

* Centefing
of a Chain.

246 «The ule of the Line of Numbers

3. Having the length and breadth of an Oblong Swperficies given in
Chains, to find the content in Acres.

It being troublefome to divide the content in Perches by 160, we
may meafure the length and breadth by chains, each chain beingq
Perchesiin length, and divided into 100 links, then will che workbe
more eafiein Arithmetick. For,
Astotothe breath in Chains:

So the length in Chains, to the content in Acres.

And thus in the former Plane A D, the breadth AC will be 7a
‘Chains 50 Links,and the leagth A B 45 Chains 75 Links ; then work-
ing as before, the content will be found as before, 34 Acres 31 parts.

4. Having the Perpendicular and Bafe of a Triangle given hi Perches, to
find the content in Acres.

If the Perpendicular go for the breadth, and the Bafe for the
Jength, the Triangle will bethe half of the Oblong, asthe Triangle
CE Disthe half of the Oblong A D, whofe content was found inthe
former Propofition. Or without halfing. d

As 3200 the Perpendicular :
So the Bafe, to the content in Acres.

- So inthe Triangle C E D,the Perpendicular being 36, and the Bale f
183, che content will be found to be about 17 Acres and 15 parts.

5. Having the Perpendicular and Bafe of aTriangle given in C bains, to
find the content in Acres. 7

As 20 to the Perpendicular :
~ Sothe Bafe, coche conrent in Acres.

And fo inthe Triangle CED, the Perpendicular EF being 7, 59,

and the Bale C D 45,75, the content will be found, as before, co be
about 17 Acres 15 parts.

6; Having

in Land mesfure, 249-

6. Having the content of a Swperficies after one kind of Perch, to find the
content of the famse Superficies, according to another kind of Perch,

As the length of the fecond Perch ,
to the length of che firft Perch : .
So the concent in Acres coa fourch number s.
and chat fourch coche concentin Acres required.

Suppofe the Plane A D meafured witha chain of 66 feet, or with a.
Perch of 16 feet anc an half, contained 34 Acres 31 parts ; and it were
demanded how many Acres it would contair, if ic were meafured
with a chain of 18 foottothe Perch: thefe kind of Propofitions are>
wrought by the backward Rule of three, after a duplicate proporti-.
on. Wherefore I extend the Compaffes fron 16, 5 unto 18, 0, and the
fame extent doth reach backward, firft from 34,31 to 31,45, and:
then from 31, 45 to 28, 84, which fhews the conrentte be 28 Acres.
$4 parts.

7..Having the plot.of a Plane with the content in Acres, to find the Scale:
by which it was plotted. | |

Suppofe the Plane A D, contained 34 Acres 34 Centefins ; if I:

_ fhoule meafure it witha Scale of ro in the inch, the length A B would

be 38 Chains, and abone 12 Centefms, and the breadth A C, 6 Chains.
and 25 Cente(ms; and the content would be found by.the third Propo-
fition of this Coaprer, to be about 22 Acres 82 parts;whereas it fhould :

| be 34. Acres 3 1 parts.

Wherefore I dividethe diftance between 23, 82and 34, 31, upon:

| the Line of Numbers, intotwo equal parts; chen fetting one foot of.
_ the Compaffes upon 10, my fuppofed Scale, I find the other to extend:
_ to 12, which is the Scale required, |

ha 8, Having the length of the Parlong, to find the breadth of ghe Acres.

i}

|

|

|
|.
{

As the Jength in Perches, to 166:
So xr Acre to the breadth in Perches.

So'the length of the Furlong being 40 Perches, the breadwh ee an;
: cre:

248 Snare The ufe of the Line of Nambers hee ae a
_ Acre will hold found to be 4. Perches. Ifthe length be so,the breadth —

for one Acre mutt be 3, 20; the breadth fortwo Acres 6,40, ©
Or if the lengeh be ‘meafured by chains.

As the length in chains unto 10:
So t Acre to his breadth ia chain meafure. qf
So the length of the Furlong being 12 Chains 50 Links the breadth _
for one Acre will be found to be 80 Links, the breadch for two Acres
I Chain 60 Links.
As 12) §0, unto 10: fo 1 unto o, 80.
Orit the length be meafured by feet meafure:
As the length in feet, unto 43560 :
So 1 Acre, to his breadth in fooe meafure.

i

So the lengch of che Beslone being 792 feet, the breadth for one
Acre will be found to be 55 feet, the breadeh for two Acres 110 ter.’

ee en ee Se

CHAP. III,

The wfe of the Line of Numbers. in folid meafure, [uch as Stone, )
: Ti imber, and the like. |

Sz c-T. I.

Of the oleate of Regular Solids.
AC REC KRiy

RRSTRRERRG PP iS fl Dag

a Cay
1 Fhaving the fide of a Square equal tothe Bafe yi any Solid gives nin
inch meafure, to find the length of a foot Solid in inch meafire.

Toon fide of a Square equal to the Bafe of a Solid, may be
found by dividing the {pace between the length and breadth

into

wae ES Le eT 1) St dota har RC Sie gel ae ae Uae MR a eld
os ? Faery ee, “penn t , r

hit in fguared. Solids) oer 24g
ae two equal patts, as in the feventh Propofition of broad mea-
dure. |

- Asthe fide of the Square in inches, to 41):57 :
eg So is x foot, to a fourth number 5
and chat fourth co the length in inches.

So in the Solid A H, the fide of the fquare equal tothe Bafe E C,
_ being about 25 inches 45 parts, the length of a foor Solid will be found
about two inches 67 patts, and the lengeh of two foot Solid 5 inches

34 parts.

As25,45, unto 41, 57 :fo1,00, unto 1, 6%
| and fo are I, 63, unto 2, 67.

2, Having the fideof a Square equal tothe Baje of any Solid given in
foot meafure, to find the length of a foot Solidin foot meafure.

As the fide of the Square in feet, unto 1:
| So is 1, unto a fourth number :
-~< And that fourth, to the length in foot meafure.

So in the Solid A H, the fide of the Square equal to the Bafe EC,
being about 2 foot 120 parts, the length of a foot Solid will be found
about 222 parts of a foot.

. As2, 120, unto 1,000: fo 1,000, unto © 471.
___ and fo are 471, unto 222.

3. Having the breadth and depth of a {quared Solid given in foot mea-
(ure, tofind the length of afoot Solid in foot mea(ure. ?

- Ag 1, unto the breadeh in foot meafure ;
So the depth in feet toa fourth number :
which is the content of the Bafe in foot meafure, Then

Asthis fourth number, untoT :
So 1, unto the length in foot meafure.

Kk So

Sie |

246 The afe of the Line of Numbers a

So jn the Solid A H, the breadch being 2 foot 5 parts, the depth
t foot 80 parts, the content of the Bafe EC will be found 4foorso —
_ parts, and the length of one foot Solid about 222 parts, the length of —
two foot Solid about 444 parts of 1000. | | |

As1,counto 2, 50 : foare 1,80 unto 4,50.
As 4, 50 unto 1,00 fo 1,000 unto 0,222.

‘s. Having the breadth and depth of a fanared Solid gives in inches, to i
find the length of a foot S olid in inch meafure.

Ast hath to the breadth in inches:
So the depth in inches toa fourth number ;
Which is the content of the Bafe in inches. Then,

Asthis fourth number unto 1728:
Sez unto the lengtli of a foot ininch meafure-

So inthe Solid AH, the breadth A C being 3° inches, and the depth
AE 21 inches 60 parts, the content of the Bafe EC willbe foundto —
be 648 inches, and the length of a foot Solid about 2inches 67 parts,

the length of cwo foot Solid 5 inches 34 parts.

As £ unto 21, 6 :{0 30 unto 648. _ -
As 648 unto 1728 : for unto 267,

Oras 12 to the breadth in inches:
So the depth in inches to a fourth number.

As this fourth number to 144:
So r unto the length of a foot Solid in inch meafure.

So in the Solid AH, the breadth being 30 inches, the depth 22 —
inches 6 parts, the fourth number willbe found tobe 54, andthe
depth of a foot Solid 2 inches 67 parts. a.

As 12 unto 21,6: fo 30 unto §4. | |
As 54unto 144.3 fo runto 2, 67. ~ |

5 H aving,

shereof given in inch mea[sre, to find the content thereof in feet.

As 41.57 to the fide of the Square in inches :
So the length in inches to a fourth number ;
and that fourth to the content in foot meafure.

- Sointhe Solid AH, the length A B being 183 inches, and the fide
_ of the Square equal co the Bafe EC about 25 inches 45 parts, the
| fourth number willbe found about 112, and che whole Solid contene

| about 68 feet 62 parts.

As 41.§7 unto 25.45 : f0 183 unte 112:
and fo are 112 unto 68, 62.

6. Having the fide of a Square equal to the Bafe of any Soltd, and the
length thereof givenin foct meafare, to find the content thereof in

feet. | |

As to the fide of the Square in foot meafure :
So the length in feet co a fourth number ;
and that fourth to the content in foot meafure,

"the content willbe found to be about 68 foot 62 parts,

As 1 unto2.12: fo ty. 25 unto 32. 35:
and fo are 32. 35 unto 68.62,

-7-Having the fide of a Square equal to the Bafe of any Solid given in
|. inch meafure, and the length of the Solid given in foot mrafure, to
| find the content thereof in fects

As12tothefide of che Square given in inches :
~ Sothe length in feet to a fourch number 5
and that fourth to the content in foot meafure.

Kk 2 Se

(70 fauare Solids pat at i ep

5.Having thé fide of a Square equal tothe Bafe of any Solid,and the length

Soin the former Solid A H; the fide of the fquare equal tothe Bafe -
_ AE, being about 2 foot 13 parts, and the length A B r5 foot 25 parte,

eo
ro

248 The wfe of the Line of Numbers —

<2 i Sa % i. ee . +. A ~ } Nise eet 1 ET eS a

F Ayw
aes
>

So inthe former Solid A H, the fide of the Square being 25 inches <
45 parts, the content will be found co be about 6& feet 62 parts. %

As t2.unto 25.45 : fols. 25 unto 32.35.
and fo are 32.35 unto 68,62.

3, Having the length breadth and depth of a [quared Solid given im 4
inches, to find the content. in inches. J

As 1 unto the breadth in inches:
So the depth in inches unto the Bafe in inches. Then,

As t unto the Bafe - 1 Aba
So the length in inches unto the Solid content in inches,

So inthe Solid AH, whofe breadth A C is 30 inches, depth A BE
21 inches, and 6 partsof 10, and lengeh A B 183, the content of the
Bafe E C will be found 648 inches, and the whole Solid content about —
118584 inches, : a:

As tunto 21. 6: fo are 30 unto 648 + :
As 1 unto 648 : fo are 183 to 118,584 : .

| : |

9. Having tbe length, breadth, and depth of a {quarea Solid given in
inches, to find the content in fect. |

Ast tothe breadth in inches + : ;
So the depth in inches to. the Bafe ia inches. |
As 1728to that Bafe - :
So the length in inches to the content in feet.

-

Soin the Solid AH, the content will be found to be about 68 fee
62 parts. o

Ast unto21.6:fo30unto 648 -
As 1728 uno 648 - fo 183 to 68,62.

Or as 12. tothe breadth in inches:
So the depth in inches to a fourth number.

rs in fauared Solids 249
As 144 to that fourth number: |
So the length in inches tothe content infeer, —

And fo alfo inthe fame Solid A H, the content will be found to be
about'68 feet 62 parts. an |

o As 12 unto 216: fo 30 unto 68.62.
As 144 unto $4: fo 183 unto 68. 62.

10.Having the length, breadsh, and depth of a fguared Solid given in
foot meafure, to find the content in feet,

{| Asx untothe breadthin foot meafure :
So the depth in feet to the Bafein feer..

As © unto that Bafe -
So the lengthin feet to the content in feet.

_ And thus inthe former Solid A H, the breadth A C will be two foot
' so parts, the depth AE, 1 foot 80 parts, andthe length AB rs fooe

25 parts; then working as before, the content of the Bafe A F will be
found 4feet 50 parts, and the whole Solid content abour 68 foot 62
parts,which of all others may very eafily betried by Arithmetick,

Ast unto2. 50 :fo I, 8ounto 4.50,
As 1 unto 4. 50: [015.25 unto 68. 62,

11, Having the breadth and depth of a {quared Solid given in inches, and
the length in foot meafure,to find the content thereof in feet.

As runto the breadth in inches :.
So the depth in inches unto a fourth number,
which isthe content of the Bafe in inches.

As 144 hath unto that fourth number : |
Sothe length in feet to.the concent in feet.

+ And fo inthe fame Solid A H, the content willbe found cobe about

68 feet 62 parts.

As.

*

250 The wfe of the Line of Nambers, Sees
As runto21.6:fo3ounto648. :
As144.unto 15. 25. fo 648 unto 68, 62.

Or as 144 unto the breadth in inches:
So the depth in inches unco a fourth number ¢
which is the content of the Bafe in feer,

As 1 hath unto that fourth number ;

So the length in feet to the conteat in feer.
And fo inthe fame Solid A H, the content will be found to be about i
68 feet 62 parts. . if

As 144 unto 21,6 - fo 30 unto 4. $e.
As I unto 4.50: [015.25 unto 68, 62.

Or as 12 unto the breadth in inches:
So the depth in inches unto a fourth number.

- As 12 unto-this fourth number -
So the length in feet to the content in feet.
And foalfo in the fame Solid AH, the content will be found to bee
about 68 feet 62 parts. A
As 12 unto 21, 6: fo 30 uato 54.
As 12 unto 54: fo 15.25 unto 68.62,

ie
Vie
if

All thefe varieties (‘and fuch like aot here mentioned ) :do follow
upon the making of the Bafe of the Solid to be E C; there would beas —
many more if any fhall begin with the Bafe E H, and fo likewife ifthey
make the Bafero be FD. >

> Tength of a foot Solid ia inches.

A Sthe Diameter in inchesunto 46 90: =
| So is 1 untoa fourth number: EB

| And that fourth to che length in inches. Z

So the Diameter of aCylinder being 15 inches, the

fourth number, will be about 3,12, and the length ofa (2=

| foot Solid 9 inches 78 parts. a
| As 15 unto 46. 90: fo 1 unto 3.129, Zz
and fo are-3. 127 unto 9. 78. Za
_- 2sHaving the Diameter of a Cylinder givenin foot mea-
fure,to find the length of afoot Solid in foot meafure, ==

| *As the Dizm ter in feet unto 1,128:
| So is 1 unto a fourth number ; aa
- and that fourth co the length in foot meafure. =
So the Diameter being 1 foot 25 parts; the Jenpth of =

_ afoot Solid will be found about 8.14 partsof 1000, Z=

Asi 25 unto1.128;fo 1.00 to 0.9027: acilin
and fo are 9027 unte 8148. ae

3. Having the Circumference of aCylinder given in inches, to find
length of a foot Solid in inch meafare. |

As the Circumference in inches to 147. 36 :
— Soisttoa fourth number;
and chat fourth to the length in inches.

W/L ELD LL ag LLL Mts dsb

Re seer aioe rr ee
id cae Bh: Mie cali Pg 252
Secr. IL

| Of the Menfuration of Cylinders.

1. Having the Disazeter of aCylinder given in inch mea/wre, to find the

the

Se

i) > ~~ Ss 6S UA TAS Poe ‘we [See Y 2°? oe ee ee ae A! aes bet eee ra ys ie vr ah os ’
tN AISA SN SARC UY ea MA AH aie a iy) ama Se YN ONT Weare eS AEN EER ME a Tt a gee ee Bebb bi es

232 she fe of the Line of Nambers
So the Circumference being 47 inches 1 3 parts,the length of a foot a

Solid will be found about 9 inches 74 parts.

é As 47. 13 unto 147, 36. fo I. 00 tO 3.12,
and fo are 3,13 unto 9.78. - | ty

Bi - 4. Having the Circumference of A Cylinder given in foot meafure, to fied -
a the length of a foot Solid in foot meafure. ; |

Asthe circumference in feetto 3.545 - y
So is 1 toa fourth number ; ,
_ and chat fourth to the length in foot meafure,

Sothe Circumference being 3 feot 927 parts, the length of a foor
Solid will be found co be about 815 parts.

7 | AS 3.927 unto 3,545 : {0 1.000 unto 0.90. 3.
rs and fo are 903 unto 815. sm F
Bc | 5 Having the fide of a Square equal tothe Bafe of a Cylinder, to find the

‘pen length of afoot Solid, i

Ca The fide of a {quare equal to the Circle,may be faund bytheeighth
mee Propofition of broad meafure, and then this Propofition may be —
wrought by the firft and fecond Propofition of Solid meafure. ‘

6. Having the Diameter of a Cylinder, and the length given in inches, to
find the congent in inches. L

>. _ Ast.128 untothe Diameter in inches:

_ So the length in inches to a fourth number;
and that fourth number to the content in inches,

Se the Diameter being 15 inches, and the length 105, the content
o: che Cylinder will be foundco beabont 18555 inches.

_ As. 1284. unto Tg : foare 105 unto 1395, 87:
and foare 1395. 87unto 18555. 34.

73 Having A

Ves

in the mesfure of Cylinders. : 253

7. Having the Diameter and length of aC ylinder in foot meafure, te find
the content in feet. | :

_ As1, 128tothe Diameter in feet:
So the length in feet to a fourth number 5
___ and that fourth co the content in feet.

So the Diameter being 1 foot 25 parts, and ehe length 8 foot and
_ 75 parts, the content of the Cylinder will be found about 10 foot 75

| parts.

| A 1,128 unto 1.25 : fo 8.75 unto 9, 69;
and foare 9.69 unto 10. 74.

8, Having the Diam of a Cylinder, ned the length givenin inches, to
find the content in feet.

As 46,90 tothe Diameter in inches -
| So the lefgth in inches to a fourth number ;
| and that fourch tothe content in feet.

| So the Diameter being 15 inches, and the length ros, the content
| will be found about 10 foot 74 parts.

| As 46.906 unto 1g: fo 105 unto 33. 58:
and fo are 3 3. §8 unto 10. 74. |

|

g. Having the Diameter of 4 Cylinder, given in inches, and the lengeb in
feet, to find the content in feet.

_ As3. 54tothe Diameter in inches -
So the lengeh in feee,to a fourch number ;

and thas fourth co the content in feet.
| Sothe Diameter being 15 inches, and the length 8 foot 75 parts, the
content will be found about 10 foot 74 parts.

| LI As
|

Midna? < oped eerie uaa VEO ee mh re al, ee Re Lee +»

2: a . Koy \ ¥ i. om we “ke SEL, sett a ny Coane hi lee ee . oe '*
RT i (ies oe a %i oy? ee aes

i 4 ‘ §

sha. The wfe of the Line of Nyinbers
“AS 13.54 Lnto Ig: fo 8.75 unto 9.69: ?
and fo are 9.69 unto 10,74. : | AL SARPARL GB

10. Having the Circumference and length of aCy! inder given ininebes to
find the content in inches. |

As 3.545 to the Circumference in inches : | a
So the length in inches to a fourth number 5.
and chat fourth to the content in inches. ,

So the Circumference being 47 inches 13 parts, and che length1o5
inches, the content will be found about 185. 55. inches. Ye

As 3.545 unto 47. 13 3 fo 1o§.unto 1396.
and fo are 1396 wnto 18555.

11, Having the Circunsference and length of a Cylinder given in inches.
to find the content in feet. nd

As147. 36 tothe Circumference in inches =

So the length in inches to afourth number 5
and that fourth co the content in feet.. he
So the Circumference being 47 inches 13 parts, and the fength 105
inches, the concent will be found about 10 foot 74 partis ‘

As 147.36 unto 47.13 ¢ fo ro¥. unto 33.53.
and{o are 33. 59 unto 1074.

BA aO re tg iret enee Aa engt of a Cylinder given in foot mean
fure,to find the content in feet. | | ;

As the 3.5.45 tothe Circumference in feet:
Sothe length in feet toa fourth number ;.
and that fourch tothe concent in fect.

So the Circumference being 3 foot 927 parts; andthe length foot
73 parts, the content will be found to be 10 foot 74 parts.
As

>

wettsy, Cie? oi ak sh Le Be) NG i Ret A A) be oe
cy ; =" pt Re E + Dy, a va% F un.

Of the menfuration of Cowes. 255
As 3. 548 unto 3, 927: fo 8. 75 unto 9,69.
and fo are 9, 69 unto 10,74. :

a 33. Having the Circumference of 2 Cylinder given in inches and the
leagma in foot meafure, to find the content in feet. |

As 42, §4. to the Circumference in inches :
bie So the length in feet toa fourth number ;
_and shat fourth to the content in feer.

So the Circumference being 47 inches 13 parts, and che length 8
foot 75 parts, the concent will be found as betore 50 foot 74 parts.

As42, §4unto 47513 :f0 8, 75 unto 9, 69: |
and foare 9, 69. unto 19.74. |

$rerv. Ii.

Of the Men[uration of Cones.

1, The Diameter of the Bafe and the length of the fide of aCone being
given, to find the [aperficial content thereof.

|

| A $7 isto233 Or 113 CO355>
~~ LAX Sois § the Diamerer 6 multiplied in 18 the fide,

, To the Superficial Content 339.29.

Soothe Diameter of the Bafe of aright Cone being 12 inches, and
the fide thereof 18 inches,the Area will be found to be 339.29. For,
If you extend the Compaffes from 7 to 22, or from 13 to 355, the
- fame extent will reach from 108, (which isthe half Diameter multipli-
ed inthe fide ) to 339. 29 the Area, or Superficial content.

2. The Diameter and Axis of a right Cone being givin, to find the Solie
Content.

As 28, |
Isto 22 ¢
, So

256 Of the Menfuration of Spheres.

So is the Square of che Diameter 144, multiplied by $ of the Axis, —
To the Solid Content of the Cone 678. 85. (wis. 86.8. —

So the Axis of a Cone being 18 inches,and the Diameter 1-2. inches,
the Solid corgent will be found to be 678.85.

\

Extend the Compatfes from 28 to 22. The fame extent will reach —

from 864 (of the Axis multiplied in. the Square of the Diameter )
to 678.85 che folid content. |

Szcr. IV.

of the Men[uration of Spheres.
1. The Diameter of a Sphere being given, to find the Superficial content.

$7 is tO 225 Or 113 (0355.
So is the Square of the Diameter 74.4
To the fuperficial content

Thusa Sphere whofe Diameter is 12 inches, the fu perficial content:
chereof will be found to be 4.52.57. :

Extend the Compaffes from 7 to 22, the fame extent will reach from.
144 (the fquare of the Diameter ) to 452. 57 the fuperficial con-
rent. |

2, The Superficies of a Sphere being given, to fied the Axt.

As 22,
Is to 7:
Sois the Superficies
To the fquare of the Diameter.

So a Sphere whofe Superficies is 452.57 inches,the Diameter there
of willbe found to be 12 inches.

Extend the Compaffes from 22to 7, the fame extent will reach from
452, §7( the Superficies ) to 144, the Square of che Diameter, chez
diftance between 144 and 1 is 12 the Diameter.

Of the Menfuretion of Primes: 257
| 3. The Axts of a Spherebeing given, to fiad the Solid contents |

As 42,
Isto 22:

So is the Cube of the Diameter
To the Solidity.

So if the Axis of a Sphere be 12 inches, the Solid content thereof
will be found to be 590. 62. 3

Extend the Compafles trom 42 to 22, the {ame extent will reach
from 1728 (the Cube of the Diameter ) to 905. 14, the Solid con-.
tent. :

4. The Solidity of Sphere being given, to find the Axis.
As 22,

Isto 42:
So-1s the Solidity

To the Cube of the Diameter, or Axis. -

So a Sphere whofe Solid content is 905. 14.: the length of the Axis

will be fonnd to be 12 inches.

Extend the Compaffes from 22t0 42; the fame extent will reach:

| from 905.14, the Solidity, to 1728 che Cube of the Axis,

Of the Menfuration-of Prifmes.

A. Prifme ss 4 Solid figure contained under Planes ; whereof the two op= -
pofite are equal, like, and Parallel 5 bus. the other are Paralelogr ams. «

Euclid, Defin.1 3. Lib.1,

1.70 find the Solid content of. a Triangular Prif me.

Suppofe a piece of Timber or Stone to be an Equilateral Triangle:

at the ends, each fide thereofbeing 2.25 foot, and the length of the .
piece .

258 “Of the Menfuration of Prifous, ae
piece 17. 75 foot, this is called ‘a Triangular Prifme. | :
1. Find the concent of the Triangle at che end of the piece ( by

the tenth aforegoing ) which will be found to be 2.19, Then fay, —

Ast,
Is tothe Areaof the Bafe:
So is the Lengthof the Piece
Tothe contentof the Piece in foot meafure.

Extend the Compaffes from 1 to-2.19, ( the Content’of the Area
of the Bafe in feec ) che fame extent will reach from i7. 75 (the

lengthof che piece in feet) to 38. 87, che concene of the Piece in
Feet. |

2. To: find the So'sd Content of a Regular Solid, whofe fides at the end |
shereof are equal, and morethan3. As 4,5, 6,7, 8, Or LOC.

Suppofe a Regular Solid, by of Timber or Pais the Plane at the

Yama e _ Bafe or end thereof being a Pentagon, or
Sn Ry iieoae piien, Figure of «5 equal fides and Angles, each
may be found exatt enough for fide being 12 inches, or one foot, and the |
thee kinds-of Menfurations length of the Solid 14 foot. | |
by taking vhe leaf diftance 1, Findthe Contencof che Bafe (or Pem
a here ro encef H tagon.) at theend, by the F. of she fecond
| Ae Section beforegoing, which will be foundto
be 1.725 foot, the Perpendicular of the Pentagon being 0 69 parts _
‘of afoor; Then fay, |

As’i, .
Isto the Content of the Bafe in feet 1.729 : |
So is the length of the piece 14 foor,
— To the Concent of the Piece in feet 24. 15. a
“‘Extend.the Compaffes from 1 to 1.725, (the Content of the Bafe))
the fame extent will reach from 14 foor, the length of che Pieceto
24.15. the Contentof the Pieceinfeer. a
And in the fame manner, if the fide of an O€tagon were 12 inches |
ort foot,the Perpendicular would be found to be 1,64,and the
length 21.§ feet,che Solidity would be found to be 103.20,
SECT. ;

SE cr. VI.

Of the Menfuration of Pyransides.

* & Piramide sa Solid figare comprehended under divers Planes, [et spon
— Bone Plane, ( which ws the Bafe of the Pyramide) and gathered together
to one Point. Euclid. Lib. Defin.12. |

_ The Bafesof Pyramids may be either Triangles, Squares, Pentagons,
Hexagons,&c.as the Prifmes were; Wherefore co meafure any Pyramis,
_ you muft firft find che Area, or Content of the Bafe, and then fay,

tp eee
—. Istothe Area or content of the Bafez. 25:

— Sois onechird part of the height ry feer,
To the Solid content 33.75 feer.. ‘

‘

_’ Suppofe a Pyramis,whofe Bafe is a Square,each fide being 18 inches, ,
| Or 1.5 feet, and the height of che fame Pyramis were 45 feer, and ic-
| were required to find the Solidity. The Area of the Bafe by the fecond
| of the fifth Se&tion beforegoing, will be found to be 2.25 feer.

__ Extend the Compaffes from 1, to 2.25.( the Content of the Bafe })
the fame extent willreach from 15. ( one third part of the height ) to
33.75. the Solid content of the Pyramid_in feer.

li And the like of any other.

|

SEC Te VII.
Of the Menfuration of Fruftums or Segments of Pyramids or Cones. .

*T "He Solidity of every Cone or Pyramid is found by multiplying
al the Areaof the Bafe ( of what form foever) into one third pare
of the Alticude ; Therefore ima Cone whofe Bafe is Circular, andthe -
‘Diameter of chat Circle is in Foot meafure 2.50,1ts Area will be found °
by what is delivered in the foregoing Seftions to be 4.91, and its. Alci- -

tude 56.25 foor; I fay, A
| Sky

266 Of the Menfuration of Fruftunes ie ee

As ft, aia)
To the Area of the Bafes 7
So isone third of the Altitude
To the Solid Content. -

Sothe Area of the Bafe of aCone or Pyramis being 4.92, andthe
A lisage 55.25, the Solid Content thereof will be found to be 92. 06

oot. - :

Exend the Compaffes from 1 to.4..91 the Area of the Bafe, che fame
extent thall reach from 18. 75, the third part of the Altitude, to
92. 06, the Solid Content of the Cone or Pyramis.

_ Baeif this Cone or Pyramis were cut off at 18 foot from the Grea-
ter end, and thenthe Leffer Bafes Area fhou!d befound to be in foot
-meafure 2.27. what fhall the Solidiey of she Fruftum be? And in this
nature do moft Timber Trees grow, and fo being cut off ought tobe
meafured, being either Squared or Growing; And no greater
Error is here committed in the Meafuring of Tim‘ ec, it being in this
form, than by the vulgar way of meafuring fuch Timber, which is, by
finding out the Square in the Middle of the P.cce, and taking of chat.
for. che true Square, but this always makes the Concent of the Piece lels
chan ie iss; The Genuine and true way is chis. |

Multiply the Area of the two Bafes together, and from the Produ&
extract che Square Root, then add this Roor, and che two Area's toge-
ther, which fum multiplied by one third part of the Lengch of the |
_ Fruftum or part fhall give the Solid Content of that piece. |

Soa Piece of Scone or Timber whofe Area atone end is 4.91. (as
in the former Piece ) and at the Smallerend 2. 27, and its lengeh 18
foot; the Solidity by the former Rule will be foand to be 63. 48
foot. For, | |

Ast, :
- Istothe Greater Bafe :
Soisthe Leffler Bafe

To a fourth Number,

Whofe Square Root being Extracted, and added to the two formet
Area’s, will produce another number. Then fay, ee

AsT>

A ee em a —
“t - 5

eae ss! A Pray a TR Th ee CRAY CS +

fs ett * Ee " mS “it, ~y , eat ph
or Segments of Pyramids or Cones. ~. 268

As f, ; |

Is co this number Iaft found : p
| So isone third of the length of the Piece,
To the Solid Content of the Piece.

- Therefore extend the Compaffes from 1 to 4. 91, the greater Bale,
the fame extent fhall reach from 2. 27, che lefler Bafe, corr. ry. A
mean Proportional between 1 and 17, Cre

15 will be found to be 3. 34, which | sp aeseens mG GT
| added to the other two Areéa’s 4. 91, Lefler R re OED

| and 2.27, Casis donein the Margin, ) Square ore 3—34
_ will produce 10.52. whichis your other Orit. 15. >

| number fought for: Then, Lom 52
___ Extend the Compaffes from 1 to 10.52, the fame will reach from 6-
| (the third pare of 18 the Length, ) to 63, 12. the Solid Content of the
Piece which is 63 foot,and halt a quarter of a foot. |
And now for Proof of this Work to be true, let us find the Solidity _
of the upper or Ieffer pare of the whole Cone which was 56. 25 foot
long. —

| The Leffer Bafe, 18 foot being cut off of the whole Length is found
| to be 2 27, and18 being taken from 56. 25 the whole length, there
will remain 38,25, and third part whereof is 12. 75. which multiplied
| by 2.27, the Bafe produceth 28.94 for che Solidiry ofthe Lefler Cone
or Pyramis, and this being added to 63.12. the Content of the Fru-
_ftum produceth 92.06. the which is equal tothe whole Cone or Py-
ee both the parts equal¢o the whole, which proveth the Work to

_berrue.

CHAP. IV.
The ufe of the Line of Numbers in Gauging of Veffels»

| He Veflels which are here meafured are fuppofed to be Cylinders,
UD or reduced unto Cylinders, by taking the mean betweén the Diz-
“meter at the Head and the Diameter at the Bongue, after the ufuai
‘Manner. ? :

Mm 4, Having

262 —- The wfe of the Line of Numbers inGangings ‘

ae Having the Diameter and the length of a Veff 1 with the Content theres
of, to find the Gauge point. ett
Extend the Compafles inthe Line of Numbers to half the diftance: !
berween the Content and the length of the Veffel, the fame extent will
reach from the Diameter to the Gauge point. ‘ :

{ put this Propofition firft, becaufe thefe kind of meafures are not.
alike in all places.

Here at London it is faid that a Wine Veffel being 66-inches in» .
length, and 38 inches the Diameter, would contain 324. Gallons,which-
if ic be crue, we may divide the fpace between 324 and 66 intotwo
equal parts, and che middle will fall about 146, and the fame extent
which reacheth from 324 to146, will reach from the Diameter ee |
nnto 17, 15, the Gavge- point for a Gallon of Wine or Oy after Lons-”

don meafure. mel Fe
The like reafon holdeth for the like meafure in all other places.

2, Having the mean Diameter, and the length of aVeffel, to find the contente.

Extend the Compaffes from the Gauge-point to the mean Diame-
ter, the fame extent being doubled, fhall give the diftance from the
length to the content.

So the mean Diameter of a Wine Veffel being 20 inches, and the

“Jength 25 inches, the Content will be found to be 34 Gallonsafter

London meafure,

For extend the Compaffes from 17.15 unto 20, the fame extent will
reach from. 23. unto 29,15, and from 29, 15 unto 34. |

In like manner, if the mean Diameter were 16 inchies,and the length
23, the Content will be found to be about 2» Gallons. ae |

For the fame extent which reacheth back from17) 15 unto 16, will
veach from 23 to 21,45, 4nd from21,.45.unto 20.

So that if the mean Diameter fhallbe r7z-inchesand rs Centefmes —
or parts of 100, the number of inches in the length of the Veffel,
will givethe number of Gallons contained in the fame Veflel: if the
Diameter fhall be more or lefs than 17) 15, the Content in Gallons
willbe accordingly more or lefs than the length in inches,

3: Having

: }
if
i
i.
ei ||
|
if

le gees

Us ee sett vs vt ds . BN Sites y ; pee se @ ; : ey.
The ufe of the Lines in Afironomy, 263

3. Having the Diameter and Content, to find the length.

| Extend the Compaffes from the Diameter to the Gauge-point, the
~ fame extent being doubled, fhall give the diftance from the Content to

the length of the Veffel.
So the giuge-point ftanding as before, if the Diameter be 38 inchesy

andthe Content 324 gallons wine-meafure, the length of the Veflel
will be found about 66 inches.

4. Having the length of a Vi effel,and the Content, to find the Diameter.

Extend the Compaffes to half the diftance between the length and

the Content, the fame extent fhall reach from che Gauge-point to the -

. Diameter. : |

So the length being 66 inches, and the Content 324 Gallons wine 4
meafure, the Gauge- point ftanding as before, the Diameter of the Vef- A

“~ fel will be found to be about 38 inches.

| C HA P. Ve :

Aftronomical Propofitions as are of ordinary afe in
the prattice of Navigation. P

| Containing fash

1, Tofind the Altitude of the San by the (hadows of a Gnomon (et Perper
diculer to the Horizon.

AA Sche partsof the fhadow,
TA are to che parts of the Gnomon : 7

So the Tangent of 45 gr.
To the Tangent of the Alticuce:

+ Extend the Compaffes in the Line of Numbers, from the parts of a
the fhadow to the parts of the Gnomon; the fame extent will give
the diftance from the Tangent of 45 gr. to che Tangent of the Suns
Altitude. . |

Sothe Gnomon being 36, and the fhadow 27, the Altitude will be k .
; Mm 2 found

me a eee

Ae i

264 «= Theufe of the Line of Sines and Tangents,

- Equinodtial Poinr, the Declination willbe found about 20 gr.

Us Ca aes hs Ni A NB te rec at Nene ae ey ek Ola ted i) SM i Lis “heh, Maia Gi A ay Te dee eh Col a Aver ae
ra st . 2 i “" - aN
.

-
-

Found to be 36 gr. 52 m. Or the Gnomon being 27, and the fhadow Lah
the Alrirude will be found to be 53 gr.8 m. Or the fhadow being 20>
andthe Gnomon g, the Alrieude will be found to be 24. gr. 14 m.asin —

the eighth Propofitionof the ule of che Tangent-line.

if the Gnomon be 22. and the fhadow 135, the Alticude is 9 gr.
35 a, as I fhewed before.

2. Having the diftance of the Suny fron the next Equint tial point, to find: 3

his declination.

As the Radius is in proportion,

co the Sine of che Suns greareft declination: wie."
So the S'ne of the Suns diftance fromthe next Equinodiil Point, .
to the Sine of the Declination required.

Extend the Compaffes in the Line of Sines,from go gr.to 23 er.30 ms
the fame extent will give the diftance from the Suns place unto his De-
clination. ‘

Sothe Sun being either in 29 gr. of Tauro, or 1gr. Of Aquarias, or

1 gr.of Leo, or 29 gr. Of Scorpio, thatis 59 gr. diltant from the next

If che Sunbe fo near the Equinoétial Point, that his Declination
fall co be under 1 gr. it may be found by the Line of Numbers. As if
the San were in 2 or. 5 m. of Ariesythat is 125 m. from the Equinocti- —
al Point, the former extent of the Compaffes from the Sine of 90 gr.
to the Sine of 23 gr. 30m. will reach in the Line of Numbersfrom
¥25.unto 50, which fhews the Declinationto be about 50m,

3. Having the Latitade of the place, and the Declination of the San, to
fied. the time of the Sunsrifing and fetting. ;

As the Costangent of the Latitude.
co the Tangent of the Suns Declination::.
Sois the Radius, Us | |
to the Stneof the Afcentional difference between.the hour of 6,
and che time of che-Suns.rifing or fétting. i

Extend the Compafles from the Tangenz of the Complement of
the Latitude, to the Tangent of the Declination: the: fame extent
wilh:

Se Da hath (t) Sent A eg OM Se aes | py ey ough Ey Sih alii ba ica
Wego ; 3 Z , -

Pe a OS ehce Sitges nh oS ai eae.
| Oa Ns ae?
will reach from che Sine of 90 deg. to the Sine of the Afcentional —
_ difference. eae atinly mah |
Orextend the Compaffes from the Co-tangent of che Latieudeto
|. the Sine of 90 gr. the fame extent will reach from the Tangent of the
Declination, to the Sine of the Afcentional difference. ©
_ Sothe Latitude being 51 gr. 30m, Northward, and the Declination
| 20 gr. che difference of Afcenfion will be found to be 29 gr, 14 me.
which refolved into hours and minutes, doth give rhour and almoft
49-m. forthe difference between the Suns rifling or fetting, and che.
hour of 6, according co the time of the year. .

Fa

4. Having the Latitude of the place, and the diftance of the Sun, fromthe.
next Equinottial pint, to find his Amplitude,

| As the Co-fine of the Latitude;

__ to the Sine of the Suns greareft Declination 3:
__ So the Sine of the place of the Sun,

| -cothe Sine of the Amplitude.

__.Sothe Latirude being 51 deg. 30m, and the place of the Sun ‘in
deg. of Aguarins,that is 59 deg. diftane from the next Equinoétial point,

| the Amplitude will be found about 3 3 deg.20 #. For extend the Com-
pafles in the Line of Sines, from 38 deg. 30 m. the Sine of the Com-_
| plement of the Latitude unto 23 deg. 30 m. the Sine of the Suns.
Pereaten Declination ; che fame extent will reach from 59 deg. unto 33
deg. 20m, Or extend them from 38 deg. 30m, unto ¥9 deg. the fame —
| extent will reach from 23 gr. 30 m. unto 33 gr.20 m, as before.

5 Having the Latitude of the place, and the Declination of the Sun, to-
find hts Amplitade.
As the Co-fine of the Latitude,
isto the Radius :
Sothe Sine of the Declination,
tothe Sine of the Amplitude,

Extend the Compa(fes from the Co-fine of the Latitude to the fine »
| Of gogr. the fame extent will reach from the Sine of the Suns Decli«

nation co the Sine of the Amplitude, | |

| Ob;

. 6 he Af sof she Line of si sines aed te

Or extend them fromthe Tangent of the Latitude ro the Sine of ced td
Declination, the fame extent will reach from che Sine of 90 gr. to the
Sine of the Amplitude. _
Sothe Latitude being 51 gr. 30 mm. and the Declinstion 2ogr. the
_ Amplitude will be found to be 33 gr.20 m. |

6, Having the Latitude of the place, and the Declination of the Sas, to
find the time when the San cometh to be due Eaft or Weft.

As the Tangent of the Latitude, |
; isto the Tangent of the Declination:
~Sothe Radius»
to the Co-fine of the hour from the Meridian.

‘Extend the Compaffes from the Tangent of the Latitude the Tan-
gent of the Declination, the fame extent will reach from the Line of —
90 gr. to the Sine of the Complement of the hour.

-Or extend them from the Tangent of the Laricude tothe Sine of oa
: gr. the fame extent will reach from the Tangent of the Declinationto
i . che Sine of the Complement of che hour;
* ‘So the Latitude being 51 gr.3© m.and the Declination 20 gr. the Sun
"a will be 73 gr. 10.m.that is 4.hours, and 53 #. trom the Meridign, weg |

he cometh to bein the Eatt or Weit.

7. Having the Latitude of the place,and the Declination of the Sun,to find f
what Altitude the Sun (hall have,when he cometh to be due Eaft or Wefhe
©
Asthe Sine of the Latitude, ri
is tothe Sine of the Declination:
So the Radius, |
to the Sine of the Altitude,

Extend the Compaffes in the Line of Sines from the Latitude to the
Sine of the. Declination, the fame extent will reach fromthe Sine of ’
4 90 gr. to the Sine of the Altitude. |
| Or extend them from the Sine of the Latitude to rhe Sine of go gr. |
the fame extent will reach from the Siae of the Declinationto the Sine f
of the Altitude.

So the Latitude being §1 gr. 30m. and the Declination 20gr. the |
Altitude will be found about 25 gr. 55 m. 8, Having

ae | in Aftronomye ) 267

8. Having ihe Latitude of the place, and the Declination of the <pn to
find what Altitude the Sun (hall have at the hour of fix.

__ Asthe Radius is in proportion,

tothe Sine of che Suns Declination s -

So the Sine of the Latitude, 7
to the Sine ofthe Altitude.

Extend. the Compafles in the Line of Sines, fromgogr. to the De= -

:
|
- clination; the fame excent will reach from the Latitudeco the Alti--
Orextend them from 90 gr. to the Latitude, the fame extent will :

_ hold fromthe Declination coche Altisude.

| aogr. the Altitudeof the Sun will be found to be about 15 g7.30 me -

‘find what Azimuth the Sun (hall have at the hoar of rece:

| AstheCo-fine of the Latitude, .

| is to the Radius ; -

| Sothe€o-tangent of the Suns Declination, . (ridian. .

tothe Tangent of the Azimuth fromthe North part of the Me- -

_ Azimuth will be found to be 77 gr. 14m. For extend the Compaffes .
inthe Line of Sines, from 38 gr. 30 m. to go gr, the fame extent will

~ reachfromthe Tangentof 70 gr.co the Tangent of 77 gr.14m-

10.Having the Latitude of the place, and the Declination of the San,and -

the Altitude of the Sun, to fiad the Azimath,

Firft, Confider the Declination of the Sun, whether it be toward ©

the North or the South,fo have you his diftance from your Pole: thea —
add this diftance, the Complement of his Altitude, and the Comple-

"ment of your Latitude, all three together, and from half the fum fub-

at che diftance from the Pole,.and note the difference,

So the Latitude being 51 gr. 30m. and the Declination of the Sun -

I. 9» Having the Latitude of the place, and the Declination of the San, to-

d Sothe Latitude being 51 gr. 30 m..and the Declination 20-¢r. the -

I ? AS-«

Fos

+
Wah

PN Sok RI RORE Ue CS ie CA en See THEIR acre es. OM SCAN I, i PNR ORL cee OEE OMe me aL mre MCane Oa Deen
z

268 —- The nfe of the Lines of Sines and Tangents,

I. As the Radius is in proportion, | ated aay
co the Co-fine ofthe ‘Alntudes | ke al
Sothe Co-fine of the Laticude, ) . Phot og
to a fourth Sine, ; ae

. 2. AsthisfourthSine,
‘2p isco the Sine of the half {ums
‘So the Sine of the difference,
toa feventh Sine.

“Then find a mean propertional between chis feventh Sine and the Ras

- dius, thismean fhall be che Sine of che Complement of half the Azi-

muth from che North partof che Meridian. f |

os Suppofethe Declination of the Sun being known by thetimeofthe _
‘ef year co be.20.gr. Southward, the Altitude above the Horizon found by i
. obfervation-1.2 gr. and the Latitude Nortiwards §1 gr. 30 ms. it were .
ae. required to find che Azimuth. | id
| The Declinationis Southward, and therefore the diftance fromthe —
Pole 110 gr, then.turning the Altitude and Latitude unto their Com- _

plements, I add them all three together, and from halfche fum fubtra& :

if the diftance from the-Pole, noting the difference after this manner : )
hs Declin. South 2ogr. om The Diftance Liogr. Om. q
je Altumde 12 re The Complement 73 O |
ae Latitude N rd Led The Complement 38. 30 )
sah The fum of allthree . .,, ;, 226). 30. Malt
; The half fum | TD 3.50) 15) ,
dl om ‘The difference 3 Is

| Thisdone, I come tothe Staff, and extend che Compafies fromthe —

“Sineot 90 gr. tothe Sine of 78 gr. and find the fame extent to reach |

from the Sine of 38 gr. 30 m. unto37 gr.30m. Or if lextendthem —
tro 1 90 gr. to 38 gr. 30 m, the fame extent doch reach from 78 gr.unto

2.7 gr. 30 m. which is the fourth Sine required. hs
Then Lextend the Compaflesagair, from this fourth Sine of 34¢r, |

es 30 m. unto the Sine of the half fum.z.13 gr. 15 m. that isto the Sine of
66 gr.45m (foralter ger. che Sméot $0 gr. doth ftand fora Sine

of

i

7 es aX, ee 22 4 Paw Tihs ee Pw 2 4 eee ae AS ee oe) Se ee), ee i e's el
bd ed ee SS ee Be + yh Sy ees, » Fe rf re)
— ee et ar of . 3

me Ct RS Ta A Rremmye ye 269
| 6f 100 grand the Sine of 7ogr. for a Sine of 100 gr. and fo the Fett
“for thofe which are their Complements to 180r. ) and tais fecond ex-
tent doth reach from theSine of the difference 3 gr. 15 m. tothe Sine.
of 4¢r. 54m. OrifI extend them from the fourth Sine of 37 gr.30 m.
ro the Sine of the difference 3 gr. 15 m. the fame exrent will reach
fromthe Sine of the half fum113 gr. 15 wm. unto 4 gr. 54m. which is
the feveath Sine required.
> Laftly, I divide the fpace between this feventh Sine of 4 gr. 54. 1.
‘and the Sine of 90 gr. into two equal parts, and I find che mean pro-
portional fide to fall on 17 gr. whofe Complement is 73 gre the double
of 72¢r. is 14.6.gr.and fuch is the Azimuth required. Riticte
Or having tound/the feventh Sine to be 4. gr. 54. I might look over
“againftit, inthe Line of Verfed Sines, and chere I fhould find 146 gr,
for the Azimuth from the North part of the Meridian; andthe Com-
| plement of 146 gr. toa Semicircle being 34g. will give the Azimuth
_ from the South part of the Meridian. - sidgrs
But if it were required to find the Azimuth in the fame Latitude of
31 gr. 30 Northward, with the fame Altitude lof 12 gr.and like Decli-
| nation of 20 gr. tothe Northward, it would be found to be only 72 gr.
| 52 m. though che manner of work be the fame as before. ;

a ___ _— — _

Declin. North. 20 gr. om. The diftance is 70 gr. Om.
Altitude 12 © The Complement 78°.
- Latitude North, 51 30 The Complement 28 30
The fum of all three 186 30 pt
The half fum 7 93 15 3
The difference 23 I§

| Here as the Radius is tothe Sine of 78 gr. fo the Sire of 38 gr.30 m,
“tothe Sine of 37 gr. 30%. which fs the tourth Sine, and che fame as
| before. sty |
| Then as this fourth Sine of. 37 gr. 30m. 1s to the Sine of 93 gr.15 m.
fotheSine of 23 gr.15 m. toche Sine of 40 gr. 20m. which isthe fe-
-yenth Sine. |

The half way between the feventh Sine and the Sine of 90 gr. doth
fallat 53 gr. 34 ™. whofe Complement is 36 gr. 26m. and thedouble -

of that is 72 or. 92 m. the Azimnch required. 4
li Na Or

i a

! i \ j
4 \ t
s : . a ‘ {bee
sal Pm Pi
ao J

¥en, ee

CIO) The wfe of the Line of Sines. } i
Or I may find thisfame Azimuth in the Line of Verfed Sines, over
againit the feventh Sine of 40 gr. 20 m. ace

11, Having the Latitude of the place, the Declination of the San, and
the Altitude of the Sun, to find the hour of the day. ;

Add the Complement of the Suns Altitude, and the diftance of the —

Sun fromthe Pole, and the Complement of your Latitude, all chree
rogether, and from half che fum {ubftract the Complement of the Al-
tisude, and note the difference, 7

7, As the Radius isin proportion
to the Sine of the Suns diftance fromthe Pole:
So the Sine of che Complement of che Latitude,
to a forth Sine.

2. As this fourth Sine,
isto the Sine of the half fum:
So rhe Sine of the difference
co a feventh Sine.

The mean proportional betweén this feventh Sine and the Sine of ©
90 gr. will be the Sine of the Complement of half she hour from the _

Meridian.
Thus in our Latitude of 51 gr.30 m, the Declination of the Sun be-

ing 20 gr. Northward, and the Altitude 12 gr. I might find the Sun to”

be 95 gr. 52. from the Meridian. ‘

Altitude 12¢r. Om, The Complement is qBgr. OMe
Declin. North 20 =o The diff. fromthe Pole yo 0
Lativude 5st 30 ~ The Complementis 38 30.
- The fum of all three 286 30
The half fum | 93 «TS
The difference Th DF

“Here as the Radius, is to the Sine of 70 gr.
So the Sine of 38 gr. 30 m. to the Sine of 35.¢7 48 ms

As

~ ail

~ zs

~ “‘Asthis Sine of 15 gr.48 m, isto the Sine of 93 gra 15 m.
So the Sineof 15 gr.15 m. to the Sine of 26 gr. 40 m.
The half way between this feventh Sine of 26 gr. 40 #. and the Sine

-of go gr. doth fall at 42 gr.4.. whofe Complement is 47 4.56 m.and
_ the double of that, 95 gr. 52 7. which converted into hours, doth give

6 hours and almoft 24. #. from the Meridian.
Or I might find thefe 95 gr. 52 m. inthe Line of Verfed Sines, over

_ againft the feventh Sine of 26 gr. 40 m.

12. Having the Azimuth, the Sans Altitude, and the Declination, to fins
the heur of the day.

As the Co fine of the Declination,
isto the Sine of the Azimuth:

So the Co-fine of the Altaude,
co the Sine of the hour.

Thus the Declination being 20 gr. Southward, the Altitude 12 gr.

and the Azimuth found by the tenth Propofition 146 gr. I-might find -

the time tobe 35 gr. 36m. that is 2 hours 22 m. from the Meridian.

2 3. Having the hour of the day, the Sans Altitude, and the Declinationy

tofind the Azimuth.

As the Co-fineof che Altitude,
isto the Sineof the hour:
' So the Co-fine of the Declination,
to the Sine of the Azimuth.

Sothe Altitude of the Sun being 12 gr. and the Declination 20 gr.

- Southward, and the Angle of the hour 35 gr. 36 I fhould find the

Azimuth tobe 34 gr. And fo it is if itbe reckoned from the South 5

but 146 gr. if it be taken from the North part of the Meridtan.

14. Having the diftance of the San from the next Eqninoltial point, te
) find his right A(cenfion.

_ Asthe Radius,
tothe Co-fine of the greateft Declination:
So the Tangent of che diftance,

To the Tangent of the right Afcen(ion.
: : Nn 3 So

The ufe of the Line of Lines in Afrommye 271

aze The nfe of the Linas of Sines and Tangents, —

Ne Pe

So the Sun being in the firft degree of eAguarius, that 1s §9 gr. die
fant from the next Equinoétial point, and the greateft Declination
23 ¢r. 30 m. the right Afcenfion will be found to be $6 gr.50 m. fhore..
of the beginning of Aries, and therefore 303 gr. 14m. :

2

15. Having the Declination of the Suz, to find his right eAfcenfion..

As the Tangent of the greareft Declination,
iscothe Tangent of the Declination given:
So the Radius | )
to the Sine of the right Afcention.

“Sa the greareft Declination being 23 gr. 30, and the Declination —

of the Sun given 20 gr. the right Afcention will be found about
56 Lre Om. | 4

rs

16. Having the Longitude and Latitude of a Star, to find the right Afcen- |
fion of ‘that Star.
17. T0 find the Declination of that Star

The ftars have little or no alteration in their Latitude, in their Lon- —

~ gitude they move forward, about 1 gr. 25 m, in an hundred years. Thefe

being krown,

As the Radius, : (points
to the Sine of the Sears Longitude fromthe next Equinotial —

Sothe Co-tangentof the ftars Latitude, :
cothe Tangentof afourth Ark.

Compare this fourth Ark, with the Ark of diftance beeweenthe
Poles of the world and of the Ecliptick. If the Longitude and Latte
tude of the Star be both alike, as when the Longitude falleth to be
among the Northern Signs, e ries, Taurtts,Gemini, Cancer, Leo, Virgoy
and the Letitude is North fromthe Eclipticks or the Longitude among ~
the Southern figns, Lilra, Scorpio, Sagitarius, Capricorn, Aquarius,
Pifces, and the Latitude Southward, then fhall the difference berween — |
ehis fourth Ark and the diftance of. Poles, be your fifth Ark. . |

Burif. the Longitude and Latitude shall be unlike, as the Longitude
in a Northern fign, and the Latitude South, or the Longitude in: a |
Southern.

ie

}

: b ap {
' ft

Southern fign, and the Latitude North, then add this fourth Ark
to the diftance of both Poles, the fum of both fhall be your fifth

Ark. And, |

As the Sine of the fourth Ark, ©
to the Sine of the fifth Ark ; ;
So the Tangent of the ftars Longitude, (nodal poine.

tothe Tangentof the ftars right Afcenfion, fromthe next Equie:.

As the Co-fine of the fourth Ark, -

to the Co-fine of the fifth Ark e--
So the Sineof the fars Latitude,

to the Sine of the ftars Declination;

:
j
| Thenfor proof of the work, if there be no former error, the pro=.-
- portion will hold,

As the Co-fine of the Latitude,
rothe Co-fine of the right Afcenfion :. -
-Sothe Co-fine of the Declination,

| tothe Co-fine of che Longitude.

of thelittle bear, which fea-mencall the Former Guard. This in ehe
year 1655, wasin7 deg. §3 m, of Leo, and fo his Longitude from the

s1-m. Northwards. Wherefore,

| AstheSineof gogr.

~~ istotheSine of §2 gr. 22 m.--

|
‘

Sois the Co-tangent of 72 gr. 51 ms -
tothe Tangent of 13 gr. 44m -

) ri Which is the fourth Ark. Then becanfe the Longitude and Latitude
4

AAS om

| in Affrommy, 273

For example, Take the upper of the two former ftars in the fquare: -

beginning of Libra 52 deg. 7 m. But his Latitude is ftillthe fame 72 gre -.

are both Northward,the difference between this fourch Ark and 23:¢r.
31 m. the diftance of both Poles will give you 9 gr. 47 m, forthe fitch .

2y
Z3
fy

pe

a7 The wfe of the Ling of Lines

As the Sineof 13 gre 44m.
cothe Sine of 9 gr, 47 m.

Sothe Tangentot 52 gr. 22 m.
tothe Tangent of 42 ¢r. 56 m.

Whichis the right Afcenfionof this ftar, from the beginning of Zi- a

bra, but 222 gr. 56 m. fromthe beginning of Aries.

Asthe Co-fineof 13 gr.44 m.
co the Co-fine of 9 gr. 47 tm.

So the Sine of 72 gre 51 m.
tothe Sine of 75 7.46 wm.

Which is the Declination of this ftar from the Fquator.

Asthe Co-fine of 72 gr. 51 m.
to rhe Co-fine of 42 gr. 56 m.

So the Co-fine of 75 gr. 46 mi

~ tothe Co-fineof 52.¢r. 7 m.

Which agreeing fo well with the Longieude of the ftar propofed is
a good proof, that the right Afcenfion and Declination were truly _

found.

Thefe are fuch Aftronomical Propofitions, as I take to be ufefulfor
Sea-men. For the firft and fecond willhelpthem cto find their Lati-
tude, chethird tofiad the Suns rifling and fétting, the4,5, 6,7) 8,9)
10,13 Prop.to find the variation of theic Compa‘s, che 11 and 12 Prop.

to find the hour of the day; and the reft toward che finding of the
hour of the nighr, For having the Latitude of the place, with the De-

clination and Alticude of any ftar, they may find the hour of the far —
from the Meridian, as in the 11 Prop, Then comparing the right ©

Afcenfion of the ftar, withthe right -Afcenfion ot the Sun, they may i

have to the hour of the night. ay
All thefe Propofitions, and fuch others, may be wroughe alfo by the —
Table of Sines aud Tangents. For where four Numbers do holdin ©
proportion; asthe firft tothe fecond, fo the third to the fourth; there
if we multiply the fecond into the third, and divide the Produét by —
the firit, the Quotient will give the fourth required, As in the

example

Bee: in Aftronitity, 279-
example of the 1$ Prop, where the Declination being given, it wasre-
quired to find the right Afcenfion. The Tangent of 20 gr. the Decli-
nation given is 3639702, which being multiplied by the Radius, the
Produ& is 36397020000000, and this divided by 4348124the Tane
| gent of 23 gr. 30 m. the Quotient is 8370741, the Sine of 56 gr. 50m,
for the right Afcenfion required.
Orif any willufe my Tables of Artificial Sines and Tangents, they
may add the fecond and third together,and from the fun fubcra&t che
firft, che remainder wil! sive che fourch required. And fo my Tangent
of 20 gr. is 9561, 0658, which being added to the Radius, mskes
19561, 0658, trom thisif they fubtraét 9638, 3019, the Tangent of
23 gr. 30 m.they fhall find the remainder to be 9922, 7639, which in
| my (anon is che Sine of 56 gr. 49 m. 56 feconds ; and fuch isthe right
_ Afcenfion required, if ig be reckoned from the next Equino@tial "aa
point. : | ae
_ The like reafon hofdeth for all other Aftronomical Propofitions, ee
asT will farther thew by thofetwo examples which I gave betore, for
the finding of the Azimuth inthe 10 Prop. becaufe they are thonghe .
_ to be harder thanthe reft, and require three operations.

In the firff Exareple,

Declin. South 20 gr. Om. The diftance 110 gr. Ome.
Altitude 1 alg sc The Complement 73° 0
Latir.North 51 30 The Complement 38 30° nt
The fum of all three 226 30 of
Thehalf fom TPF UES we
The difference 3 IS.

The firft operation will be to find the fourth Sine; and that isdone -
by adding the Sine of the Complement of the Altitude tothe Sine of
the Complement of the Latitude, and fubtracting the Radius: fo ad-:
ding 9990, 4044 the Sine of 78 gr, unto 9794, 1495 the Sine of 38 gr.
30 m. ihe fum will be 19784, 5539. And the Radius being fubtractéd,
the remainder 9784, 5539 is the fourth Sine, and belongeth to -
37 £7.39 Mm. } ; 4

The fecond operation will be to find the feventh Sine, and a is _*
) one. - i

Ue St ON Cee "eS ETN: NCAR, Be eT Sea RIT eae aT ee Re Ae ee

f 3K

Cle ior The ufeof- thelane of Lines.
a. done by adding the fine of the half fumto che fine of the difference, —
ae and fubtracting the fourth fine. So the half fum being 113 gr. 15. I
- take his Complement toa Semi«circle,and fo find his fine to be 9663, —
2168,to which I add 8753, 5278, the fine of the difference 3 gr.i15 mm,
and the fumis 18716, 7446. Fromthis Itakethe fourth fine 9784, —
5539, and the remainder will be 393231907, which is the feventh line,
and belongeth to 4 gr.5 4 m. ; a 4
The third operation willbe to find the meaa proportional fine be- |
tween the feventh fneand the Radius. This in common Arithmetick is
done by multiplying thetwoextreams, and taking the {quare rootof |
theProduct. Asin findinga mean proportional between 4 and 9, we
multiply 4intog, and the Product is 36, whofe fquare root is 6, the
mean preporticnal between 4 and g. But here it is done by adding the _
fine and the Radius,- and taking the half of them. So the fumof the _
laft feventh fine and the Radius is 18932, 1907, and thehalf of that —
9466, 0953, which is the mean proportional fine required, and be-
longethto 17 gr. whofe Complement is 73 gr. and the double of thag_
146 gr. the fame Azimuth as before. j

Inthe fecond Example. | . |
“Declin. North 20¢gr. om. The diftance 7O gr. Om)
eitiendes ro rere TheComplement «78 200
‘Latitude N. = 5t 30. TkeComplement ~ 38 430. |
ie oo The fum of all three 186 30 i
a | ea ant —
Bis ee | The half fum 93 15
i. The difference 23 159% |

The firft operation will be to find the fourth fine, and that is here |
es 9734, $539, as inthe former Example. | | K |
_. The fecond operation will be to find the feventh fine; andfo here
‘ the fine of the half tum 93 gr. 15 m, being the fame with che fine of
86 gr.45 m.his Complement to 180¢r. I find it to be 9999, 3009, 60
which Ladd. 9596, 3153, the fine of the difference 23 gr.15 m.and |
the fum is 19595, 6162. From this I cake the fourth fine 9784, 5539)
‘> and the remainder willbe 9811, 0623 for the feventh fine, and be |
Jongeth to 4ogr. 20 wm,

The |

| gy Bs Sa cat elie
) er ai y

Nige'

The ufe of the Line of Sines and Tangentss. 277
The third operation will be to find the mean proportional Sine be-

tween the feventh Sine-and the Radius. And fo here the Radius being
added tothe feventh Sine, the fum will be 19811, 0623, and the halfof

that 9905, §311, doth give the mean proportional Sine belonging to.
| about $3 gr. 34. m. whofe Complementis 36 gr, 26, and the double
_ of that 72 gr. 52 m.the fame Azimuth as befere.

T have fee down thefe three Examples thus particularly, that I mighe

fhew the agreement betweenthe Staffe and the Canon. But otherwife I

_ might deliver both the Precept andthe Work, for the two laft, more“

_compendioufly. For generally in all Spherical Triangles, where three

|
|

‘this-manner,

fides are known, and an Angle required, make that fide whjch is oppo-
fite to the Angle required, to bethe Bafe; and gather the {um, the half

‘fun, and che difference as before,

Asthe Rectangle contained under the Sines of the fides,
is to the Square of the whole Sine: . (difference,
So the Reétangle contained under the Sines of the half fam and the
co the fquare of the Co-fine of the half of the Angel.

Then for the work, we may for the moft part leave out the ewo laft
figures; andif they be about 50, pue an unite to the fixth place, after

The fecond Example.

7ogr. OM
78 i) 9990 | 40
38 30 9794 | I§

186 30 19784] 55

eee ee

ia 5 aden 9999 | 30
| 23 «15 9596 | 32
20000 | CO
ee en ens |
39595 | 62
19811 | 07 |
360 26 9905 | $3 $3 fre 34.
72 <4 §2 109 8

| cbr Oo Or

aS Tha nfo of the Lines, of Sines and. Tangents. ae
Or for fect: Numbers as are to befubtraéted, I may takethem ous
of the Radius, and write down the relidue, and.then add them toge-
cher with thereih. Asinche fame fecond Example, the Sines.of 78 gr.
and of 38 gr. 30m. being the Numbers to be fubrragted; if Itake —
9990, 4044. the Sine of 78:gr. ourof the Radius 10000, 0900; the
relidueis 9. 59563 and fo the refidue of 9794, 1495 16205-8505.
Wherefore inftead of fubtracting thefe Sings, 1 may add thefe refidues
atrer chigs manner: |

79 fire © Mm.

7 ° 9| 59
3 30 205 | 85
186 30 .
ELS 9999 | 3°
ta a ge 9596 | 32
e : 19811 | C6 : |
36 + 26 9905 ee 53 Ore 34 Me |
as wins u tO 4

Having thefe means to
find the Suns Azimuth, we —
may compare it with the ~
Magnetical Azimuth, and
fo find the variation of the —
Needle. 1

For let the Circle
AMB, drawnby the Cen-_
ter Z, be aPlane, parallel —
to the Horizon; A the |
Point whereon. the Sun |
bears from us, M the North —
point of the Mapnetical
Needle, and the Angle

‘AZM, the Magnetical |

. Sy By 8 -» 5 Azimuth, If.we find the»
Suns Azimuth as before, tobe 72 gr. 52 ™. from-the North to the”
| Weftward) -

| ws
———S e

= ————————

| of my Friends, dra
| ter, and two Needles, the one above 6 inches, and the other 10 inches
long, where I made the Semidiameter of my Horizontal Plane A Z

ER Rae he Oe ae ML Pe et oy Cee ee

}

thenfe of the Lines-of Sines avd Tangent; 239

‘Weftward, we may allow fo many gr. from AuntaN; and fo we hav?
‘the true North point of the Meridian, and confequently, the Eall,
South, and Weft Poiats of the Horizon, and the dittance becween N
~and M fhall be the variationof the Needle. Sothatif the Magnetical

Avimath AZM thall be 84gr.7. andthe Suns Azimuch AZ N
72 9re 52m then mutt N Z M the difference between the two Meri-

| dians, give the variation to be 11 gr. 1§ tm. 25 Mr. Borough heretofore -
found it by his obfervations at Linsehonfe inthe year1g8o0, But if the

Mapnetical Azimuth Z M fhall be 79 gr. 7 . and the Suns Azimuth

AZN 7247.53 m.- then fhall the variation N Z M be only 6 gr. 15 #7.

as Lhave {ometimes found it of late, Hereupon enquired after che
place where Mr. Boreugh obferved, and went to Limehoufe with fome
and took with us a Quadrant of three foot Semidiame-

19 inches: and towards night the 13 of Fune 1622, I made obfervatioa
in feveral parts of the sround, and found as followeth : °

=

Al, O|AZM.LAZN.| Variat. [)

peg USE

Gre Wf. | Gr. tat.) GRA: Gr. M,
19 oe} 82 4AVAP F216 IO
18 5 | 80 50 | 74-44 .6 6

| U7 ipdd WES Pd Lh Se boy aoe

1% 05799 B12 76-95 55 |

| 16 18 | a8 ro 72°32 Be a dea

116 0177 “50,72 1815 40 |

eR A Ry 2 les 49 ¢ 3
d9 29170 12164 25}5 471)

eal J Dds Sage ek a ee eh cen emerge RA al er amma

@

¥ Oo 2 CHAP}

- 4 Le. i -~) ae
; -
& ~

| The nfe of the Line of Sinesand Tangents, |

S

ee

atte . . |
a Containing fuch nautical queftions, as are of ordinary, —
hs ufe, concerning Longitude, Latitude, Rumsbs. |

and Diftaxce.
1. Tokeep an account of ihe Ships way.

|
i
i
ah He way that che Ship maketh, may be known to am
Ro te old Sea‘man by experience, by others it may be.
1 found for: fome fall proportion of time, either by the
! Log Line, or by the diftance of two known marks on the:
7 Sips fide.
The time in which it maketh this way,may be meafured
3 || -bya Watch, or by a Glafs, or by the Pulfe, or by re-_
peating a certain number of words. Then as long as.
the wind continuech at chefame ftay, it followeth by pro-
portion, :
As the time given, isto an hour:
So the way made, ta an hours way.

Suppofe the time to be 15 feconds, which makea quar=
ter of aminute, andthe way of the Ship 88 feet: chen —
becaufe there are 3600 fecondsin anhour, I may extend _
the Compaffes in the Line of Numbers, from 15 unto

- 3600, and the fame extent will reach from 88 unto. \
21120. Or Imay extend them fromi5 unto 88, andthis _
i ad extent will reach from 3600,unto 21120, according to
cA the ordinary work in Arithmetick, |
fis : As 15, unto 3600:
a So 88, unto 21120.
Which fhews that an hours way came to 21120 feet.
‘|; _ Bue this werean unneceflary bulinefs,to hearken afrer-
Et feecor fathoms. Ie fufficeth ourSea-men tofindthe way
ay . Hae Fe :
|| oftheir Ship in Leagues or Miles. :
And they fay that there are 5 feet in apace, 1000 paces
ina Mile,and 60 miles in.a degree, and therefore 000
| eee

The ufe of the Line of Sines and Tangents: 281

feet ina degree, Yet comparing feveral obfervations, and their mea-

fures with our feet ufual about London, I find that we may allow
352000 feet to a degree; and then if I extend the Compafies in the
Line of Numbers from 352000 unto 21120, I fhall find the fame ex-

| .tent to reach from 20 Leagues, the meafure of one degree, to 1,2,and

from 60 miles 03, 6, according to Arithmetick, which fhews the hours

way tobe league, and 2 tenthsof a league, or 3 miles and 6 tenths of

a mile.
AS 352000, unto 21120,:
SO 20, 00, unto 1, 20,
_ and 60, 00, unto 3, 60,

__ But to avoid thefe fractions, and other tedious reduétions,! fuppofé -
it would be much berter to keep this account of the Ships way (as

alfo of the difference of Latitude, and the difference of Longitude )

by deg. and parts of deg. allowing in 100 parts to each deg. which we -
_ may therefore call by the name of Cente/ms. For fo doing there would

_ befome agreement between the account and the days fayling. Ordina-
_tily the fhip goesa degree ina day, as it may appear by comparing fe-

_ Veral Journals co the Eaft and Welt Indies. The time of paflage bee .
_ tweenthe Lizard and the Southermoft. Cape of frica,.is commonly
- faid tobe abour 3 months, and the diftance is not much different from _
90 degrees. ‘

_ Again, this sccount by degrees and Centefms would be more exacts .
and the addition, fubtra@ion, multiplication, divifion of them more.
_¢afie. Neither would this be-hard toconceive, For, :

Cente (uss, Minwtes, Leagues. -
If 100 do equal. 60 and 209, : ha
then so fhall equal 30 and 10,
and 5. beequal. 3 and I.

And fo inthe former example of 82 feet in 15 feconds, having firft :

found that the hours way is-about 21120 feet.

If I extend the Compaffes from 352000, unto 21120, as before, .
IT fhall find the fame extent to reach from 100 unto 6, as before, which
thews that the hours way required is 6 Cent, fuch as 100 domakea .

degree, and 5.do make an ordinary league.
__ This might alfo be done at one operation. . For upon. thefe fuppofi-

Uons, divide 44. feet into4s lengths, and fer as many of them as you -
May conveniently between twomarks onthe fhips fide, and note the -
Aeconds of the time in which the ship goeth thefe lengths, fo the pro- .
portionwillhold, | | EM sh | Ag. .

*

Pe Te Om ot W nS
VRP ais ge ot ee
a

28: , the fe of the Lines of. sines and Tove

Bet As the feconds, tothelengths: ay
a So 1 hour, unto the Genteims. ry
The lengths divided by the time, fhall give the Cent. “which the tip,
pis goech in an hour.

is Suppofe the diftance between the two marks to be 60 lengeths(which —

are 58 feetand S inches ) and let the time be 12 feconds : extend the
i Compafles from 12to 1, inthe Line of Numbers ; fo the fame excent |
Fs will reach from 60 onto 5. Orextend them from 12 unto 60, andthe |
fame extent will reach from 1 unto 5. This fhews that the (hips way
_ is according to § Cent. in an hour. |
Tais may be found yet more ealily,if the Log-line fhall be fitted to

; the time, Asif the time be 45 feconds, the Log-line may havea knocat
the end of every 4.4 feet;then doth the fhip run fo manyCent.inan hour —
as there are knots vered out in the {pace of 45 feconds.If 30 feeonds do
{gem to bea more convenient time, the Log-line may havea knot atthe —
end of every 29 feet and 4 inches; and then alfo the Cent. will be as ,
many asthe knots: Or if the knots be made to any fet number of feet,
the time may be fitted unto the diftance. As if the knots be made at |
clte end of every 24 feer, the Glafs may be made 24 feconds,and fome- _
what more thanan half of a fecond, and fo thefe knots will fhew the |
Cent. If there be 5 knots vered out ina Glafs, then 5 Cent. if 6 knots, |

~~ thea the fhip goeth 6 Cent.inthe {pace of an hour, and fo inthe ref
«For upon this fuppofition, the proportion between the time and the |
feet willbeas45 unto 44. Bur according to the common oppo
ait rt fhould feem to be as 45 unto 372, or ia lefler cerms, as 6 unto §.
or Thofe which are upon the place may make proof of both, al
~ follow that which agrees belt with their experience.

2. By the Latitude and differ ence of i Spore to find the diftance ‘m8 |

; | eourfe of Eaft and Weft.
.. _ Asthe Sine of 90 gr.
to the Co-fine of the banitulie: :
FS So the difference of Longitude atthe Equator,

co the diftance required on the parallel. |
)
i Extend the Compatffes fromthe Sineof go Pa unto the Sine of the’
di: Complement of the Latitude ; the fame excent thall reach in the Line.
| ae of Nambers, fromehe difference of Longitude to the diftance. *

me | -

4 irl. f

a my i

mee ; * 4 if
Ke |
‘Re / :

.
’
ae

|

att Ge eee Pe en be (4
oe

The ip of the Lines or Sines and aac ‘ 283

So the meafure of one degree in the-Equator being 100 Cente the
“diftence belonging to one deg. of Longitude in the Latitude of §1 gr.
30 m, Will be found about 62 Cent. and 4.

Or if the meafure of a degree be 60 miles, che diftance will be
found abous 37 miles and}. If the meafure be 2oLeagues, then

eee Frnp, ho

aid
Cer

“fad ©

ee)
TD Te TT

§ 2)

PTT

5S!

1 el Ba ae

| i= |

ose I 2 fc | ee CIE 7]
| we 1,”

|

|

almoft 12 a baaites and 2, if the meafure be 174, as inthe Spanifh

| Charts, then fomewhat lefs than 11 Leagues failing upon this parallel,
_will givean alteration of one degree of Longitude, .

3. By the Latitude and diftance npon a conrfe of He or Wf to find the
difference of Longitude,

If the diftance be given in Leagues or Miles, reduce them into Cen-
' telms, then will the proportion hold.

byt | | As

caso 4%
ee
a3 a

~ of the fixth Rumbto the Meridian. Bue this Rumb fo found

284 The wfe of the Lines of —
~ Astle Co- fine of the Latitude, - |

to the Sineof 90 gr
' Sothe diftance on the Parallel

to the difference of Longitude.

Extend the Compaffes from the Sine of the Complement of che La-
titude, to the Sine of 90 gr. the fame extent will reach in the kine of
Numbers from the diftance to the difference of Longitude. |

So the diftance upena courfe of Eaftor Weft, in the Latitude of
SI gr. 30 m, being 100 Cent. the difference of Longitude willbe
found 1.60, which make one degree and 60 Cent. or 1 Lr. 36m |

Or if it be 6o miles, the difference of Longitude will be 96,which —
alfo maker gr. 36 m.as before, |

4. The Longitude and-Latitude of two places being given, to find the-Rumb '
leading from the one tothe other. Tok i

Asthe difference of Latitude, rie
to the difference of Longitude:

So the Tangent of 45 gr,
tothe Tangent of thecommon Rumb.

Extend the Compaffes inthe Line of Numbers from the difference _
of Latitudes to the difference of Longitudess the fame extent will _
give the diftance fromthe Tangent of 45 gr. unte the Tangentofthe —
Rumb, according to the Proje&ion of the Common Sea-chart. ie |

So the Latitude of the firft place being so gr. the Latieude of the
fecond §2¢r. 30 m, andthe difference of Longitude 6 gr. the Rumb —
will be found to be about. 67 gr. 23 mm. which is near the inclination |
is always
greater than ic fhould be, and therefore to be limited; which may be
‘done fufficiently for the Sea-mans ufe, after this manner: |

As the Sine of 90 gr. |

to the Co-fine ot the middle Latitude:
“So the Tangent of the common Rumb

to the Tangent of the Rumb required.

M. Extend |

~

3 ‘Sines and Tangents in Navegation, ss alee
Extend che Compaffes either from the Sine of 90 gr. unto the Sine
of the Complement of the middle Latitude, the fame extent willreach —
from the Tangent of the Rumb before found, uato the Tangent of the
_ Rumblimited. * | |

Or elfe extend them from the Sine of 90 gr. unto the Tangent of the
Rumb before found ; the fame extent will reach from the Sine of the
pence of the middle Latitude, unto che Tangent of the Rumb
_ fimited.

- So the middle Latitude between 50 gr. and 52 gr. 30 m. being 51 gr.

15 m. and the Rumb before found 67 ¢r. 23 m. the Rumb limited will
_ be found to be about §6 gr. 20 #. which is but 5 m. more than the in-
_clination of the fifth Ramb to the Meridian. |

If any pleafe co work by the Canon, he may joyn both thefe in one
| Operation.

Asthe difference of Laticude,
corhe difference of Longitude : ‘.

So the Co-fine of the middle Latitude, i
co the Tangent of the Rumb required. i

>. This Rumb may be found by the help of the Aferidias Line upon ©
the Staff, For if Itake the difference of Latitude out of the ALcridias
Line from 50 gr. unto 52 ¢r. 30m. and meafure it in his Equinodtial,
or atthe beginning of the AMderidian Line, I fhall find ic there to be
equal to 4 gr. which may be called che difterence of the Latitude in-
larged. Wherefore I work as if she difference of Latitude were 4 gr.

Asthe difference of Latitude inlarged,
tothe difference of Longitude :

So the Tangent of 45 gr.
to the Tangent of the Rumb required.

And extend the Compattes inthe Lineof Numbers from 4 unto 6:
fo fhallI find the fame extent to reach from the Tangent of 45 gr-
unto the Tangent of 56 gr. 20. and this is the inclination of the

Rumb required.

de 6.By

286 The ule of the Lints of

L oe

6. By the Ruwsh and both Latitudes, to find the diftance upon the Rurabe i

As the'Co- fine of the Rumb from the Meridian, ra
tothe Sine of 9° g7s ihe ts
So the difference between both Latitudes, | -
to the diftance upon the Rumb.

Extend the Compafies from the Sine of the Complement of the
Rumb,unto the Sine of 90g7. the fame extent in the Line of Numbers
Shallreach from the difference of Latitude unto the diftance upon
the Rumb. | arta: “

So the Latitude of the firit place being 50 gr. the Latitude of the -
fecond 52 gr. 30 ws. and the Rumb the fifth fromthe Meridian, IE T
extend the Compailes from33 gr. 45 ™. unto the Sine of 90gr. I fhall —
find the fame extent inthe Line of Numbers co reach fron 2 gr. 50.
Cent. t04.gr. 50 Cent. and {uch is the diftance required. p

7. By the diftance and both Latitudes to-find the Rumbs

Asthe diftance onthe Ramb,
co the difference between both Latitudes :

So the Sine of 90 gr.
to the Co-fine of the Rumb from the Meridian,

+ me

-
aot
ee ST

hy

a os

Extend the Compaffes in the Line of Numbers from the diftance |
uatothe difference of Latitudes ; the fame extent will reach in the
Line of Sines from go gr. unto the Complement of the Rumb. _ a

So the one place being in the Laticude of 50 gr. the other in che ©
Latitude of 52 gr. 30%. and the diftance between them 4 gr. $° Cette
1f Lextend the Compaffes from 4.50 unto 2. 50 in the Line of Num-—
bers, I: fhall find the fame extent to reach from the Sine of 90 gr.unto
ehe Complement of 56 gr. 15 wm. and {uch is. the inclination of the |
Rumb required. — srsrinr i waive sing) arly be on |

body

|

a 1
}
|
|

8. By

Ae HE Be A Ee ed Bre 2 Be By BE
¢ a i ec F Ee +.) SO Se ae Cate ys at OM tates hcl: f or.
i rer ee = a es Pea ee Fae . te eee ee wae aah 8

beg ‘Sines and Taygents 1 Navigation, 284

| 8. Byone Latitude, Rumb,and diftance,to find the difference of Latitudes.
As the Sine of go gr. |

tothe Co-fineot the Rumbfrom the Meridian:

Sothe diftance upon the Rumb,
to the difference between both Latitudes.

__ Extend the Compaffes in the Line of Sines, from 90 gr. unto the
_ Complement of the Rumb, the fame extent in the Line of Numbers,
_ will reach from the diftance, unto the difference of Latitudes. |
Sothe leffer Latitude being sogr- and the diftance 4 gr. 50 Cents
| upon the fifth Rumb from the Meridian: If I extend che Compafies
from the Sine of 90 gr. to 33 gr. 45 #. I fhall find che fame extent to
reach from 4, 50 in the Line ef Numbers unte 2-50; and therefore

| the fecond Latitude to be 52 gr. 30 m7. }

9.Bythe Rumb and both Latitudes, tofind the difference of Longit#de.

Asthe Tangent of 45 gr.
to the Tangent of the Rumb from the Meridian -

So the difference of Latitudes. |
to the difference of Longitude in the common Sea- chart.

.

| ~ Extend the Compaffes from the Tangent of 45 gr. unto the Tangent

of the Remb 5 the fame extent will reach in the Line of Numbers from

| the difference of Latitudes unto che difference of Longitude, accord-
| ingto the Projeation of the Common Sea-chart. |
| So the firft Larieude being 50 gr. and the fecond 52 gr. 30 m. and the
| Rumb the fifth from the Meridian : if Textend the ompaffes from
| the Tangent of 45 gr- unto § Gor. 15m. 1 fhall find clf@fame extent to
| reach from 2. 50 in the Line of Numbers to be about 3. 75, which
make 3 g7.4.5 %. But this difference of Longitude fo tound, isalways
leffer than it fhould bes and therefore to be enlarged, which may be
done fufficiently for the Sea-mens ufe after this manaer -

| Asthe Co-fine of the middle Latitude, —
tothe Sine of 90 pr. —
So the difference of Longitude in the common Seaschart,

a, tothe difference of Longitude inlarged. :
r | Ppz Extend

A .
"shal “es a
pe Cae <3 Sri

238 The ufe of the Lines of ae
Extend the Compaffes from the Sine of the Complement. of the-
middle Latitude, unto the Sine of 90 gr. the fame will reach inthe —

‘Line of Numbers from the difference of Longitude before found, unto

the difference of Longitude inlarged. |
So the miedie Latitude in this example being 51 gr. 15 . and the |
difference of Longitude before found, 3 gr.7 5 Cente che difference of —
Longitude inlarged will be found abours gr. 99 (ent. which are
near 6 gr. . | . ;
If any pleafe to work by the Canon, he may joyn both thefe in one
Operation. :

As the Co-fine of the middle Latitude,

to the Tangent of the Rumb from the Meridian:
Sothe difference of Latitude,

tothe difference of Longitude required.

2. This difference of Longitude may be found by help of the Me-
ridian Line upon the Staff. For if 1 take the proper difference of La-
titude out of the Meridian Line, and meafure it in his Equinoétial, or
at the beginning of the Meridian Line,l fhall find the Latitude inlarged
to be equal co four of shofe degrees.

Asthe Tangent of 45 gr. .

to the Tangent of the Rumb from the Meridian <
'. So the difference of Latitude inlarged, 7 ,
co the difference of Longitude required. tg

Wherefore having extended the Compaffes, as before, from the
Tangent of 4 Sgr. unto the Tangentof 56 gr. 15. the fame extent |
will reach from 400 in the Line of Numbers, unto §.99, which fhews
the differenceof Longitude to be abouts gr. 99 Cent. or about halfa
minute fhort of 6 degrees.

10. By the Rumb and both Latitudes, to find the diftance belonging to the
’ Chart of Mercators Projection. i a |

Takethe proper difference of Latitudes out of the Meridian Line 1
of the Chart, and meafure itin his Equinoétial, or one of the Parallels, _
and it will there give the difference of Latitude inlarged« |

As

Asthe Co-fine of che Rumb from the Meridian,
tothe Co-fineof 9o0¢r,

So the difference between both Latitudes,
to the diftance upon the Rumb.

‘ Then extend the Compaffes from the Sine of the Complement of
_ the Rumb unto the Sine of 90 gr. the fame extent will reech inthe

_ Line of Numbers, from the Latitude inlarged, unto the diftance re- -

_ quired. Or extend them from the Complement of the Rumb tothe
_ Latitude inlarged, the fame extent will reach from 90 gr. unto the
| diftance.

For example, Let the place given be A, in the Latieude of 50 gr.D,
inthe Latitude of 52 ¢r.30m. AM thedifference of Latitudes, and
the Rumb M A D the fifth from the Meridian. Firft, f take out A M,
_ the difference of Latitudes, and meafare itin A E, one of the Parallels.
_of the Equinoétial ; I find itto be very near 4 gr. this is the difference

Of Laticudesinlarged. Then if I extend the Compaffes from the Sine

| Of 33 gr. 45 m. the Complement of the fifth Rumb,unto the Sine go gr.
“unto 7.20, And this isthe diftance belonging tothe Chart. Where-
fore I take out thefe 7 gr. 20 Cent. out of the Scale of the Parallel
_ AE, and prick it down upon the _Rumb from AuntoD, where ie
_meeteth with the Parallel of the fecond Latitude. Laftly, I meafure it in
the Meridian Line, fetting one foot of the Compaffes as much below
_ the leffer Laticude, as the other above the greater Latitude, and find ie
‘tobe 4 gr. 50 Ceat. which is the fame diftance that I found before ia

the 5 Prop. en

11. By the way of the fhip, and two Angles of pofition, to find the difPance-
between the Ship and the Land.

The way of the Ship may be known as inthe firft Prop. The Angles

may be obferved either by the Staff, or by a Needle fet on the Staff.

For example, fuppofe that being at A, I had fight of the Land at B,

the Ship going Eaft Northeaft from A toward C, and the Angle of

the Ships Pofition B AC being 4.3 gr. 20 m. and afrer thatthe Ship

had made 10 (ent. or two Leagues of way from A unto sr obs
aes rve

Sines and Tangentsin Navigation, Bg

_ I thall find the fame extent co reach inthe Line of Numbers, from goo

to 33 3, and fuch is the diftance beeween A and B, and it reacheth from
‘43 gr.20 m. unto 27 inthe Line of N
‘from Deo B.

~

“ago whe wfe of the Lin of
ferved again, and found the fecond Angle of the Ships PofitionB DC

~~

to be 58 deg. or the inward Angle BD A, tobe 112 deg. chen may I
find the third Angle ABD, tobe 14d

eg. 40m, Gither by Subtraction,
_ or by Complement unto 180 gr. |

a

5

BHOA GERBER ome

\
ee
*
g®
et
at
os
aoe
pane
cuenneneeess

¢GSEE08 Se eoscoe

4
a

i)

‘In this and the like cafes,1 havea right Line Triangle, in which there |
is one fide and three Angles known, and it is re

quired ¢o find the other
‘wo fides, and the Canon for it is this ; . ie |
As the Sine of an Angle oppofire to the known fide, ei
isto that known fide :
“So the Sine of the Angle oppofite to the lide required,
is to the fide required. |

Wherefore I extend the Compaffes froin 14 gr. 40m. inthe Sines,
‘to rointhe Line of Numbers, and this.extent doth reach from 58 gr

umberss and fuch isthe diftance
Thefe two diftances being known, I may fet out the band npod ag
fires: | Charts

oe

|

i
I

|
|
|
|
|
\

See a
a

-

Point B, thall here refemble the Land required,

12, By knowing the diftance between two places on the Land, and how they -
bear one from the other, and having the Angles of Pofition at the Ship, .

tofind the diftance between the Ship and the Land.

If ig may be conveniently, let the Angle of Pofition be obferved

at uch time as the Ship cometh tobe right over againft one of the
| pecs! Asif the places be Eaft and Weft, feekto bringone of them —

outh or North from you, and then.obferve the Angle of Pofition, fo

fhall you have a right Line Triangle, with one fide and three Angies, .
whereby to find the two other fides. Firit, you have the Angle :

or Pofition at’ the Ship, then a right Angle at the place that is over

_ againft you, and the third Angle at che other placeis the Complement
tothe Angle of Pofition. Wherefore,

As the Sine of the Angle of Pofition,
is to the diftance betweenthe two places: -
-Sothe Co-fine of the Angle of Pofition, . ) |
cothe diftance between the Ship and che nearer place.»
And fo is the Sine of 90 gr.
tothe diftance from the Ship to the farther place: .

and the Angle of Pofition 29 gr. the nearer diftance willbe. found

about.72 Cent.and the further diftance about 31 Cent.
Or howfoever the Angle of Pofition were obferved, the diftance ~
between the Ship and the Land may be found generally as in this -
example: - aM brid: :

~. Suppofe A and D were two head Lands knownto be Eaft Northeaft, .
and Weft Southwelft, 10 Cent. or two Leagues one fromthe other ;

and that the Ship being at B, I obferved che Angle of the Ships Poli-
tionD B A, and found itto be 14 gr. 40m, and that D did bear 9 gr.
30 mand A 24.97. 10%. from the Meridian BS, thisexample would
belike the former. For if the Angle SBD be ogr. 30m. fromthe
South to the Weftward, then fhall NDB be ogr. 30 m. Hee ihe
" ort

Sines and Tangents in Navigation. - 20t
| Chart. For having fet down the way of the Ship, from Ato D, by
that which I fhewed before in the ufe.of the Meridian Line, I may by
the fame reafon fet off the diftance A B and D B,which meeting in the.

Sothe places being 1§ Cent. or three Leagues one from the other,” .

452 0—st«*«s He aft of the Limes of

-Northtothe Eatward. Take thefe 9 gr.30 ws,out of che Angle N D E,

‘ pei :
a
a

v
=
sy
ite
.

which is 67 gr. 30 m. becaufe the two head Lands lie Eaft Northeaft, ;

and there will remain 58 gr. for the Angle B DE, and the inward An:
eleB D Aoutof 180gr. Take thefe two Angles ABD and BDA
outof 180 gr. and there will remain 43 gr, 20m. for the third Angle
“BAD. Wherefore here alfo arethree Angles and one fide, by which
L.may find the ewoother fides, as ia the laft Prop.
- Thefe Propofitions thus wrought by the Sraft, are fuch as I thoughe

to beufeful for Sea-men, and chofe that are skilful may apply the ex-

ample to many others. Thofe that begio, and are willing to practice,
may bufie chemfelves with this which followeth.

Suppofe four Ports, L,N,O,P, of which L is in the Latitude of
so gre N is North from L 200 Leagues or 1000 (ent. O Welt from
L roo Cent. and P Weft from N 1000 Cene. fo that L and O will be
in the fame Latitude of 50 gr..N and P both in the Latitude of 60 gre
Then let two Ships depart from L, che one to touch at O, she other at
N, and then both to meet at P, thereto Lade, and from thence tore:

diffs Long:2%0

i
Pee eis eesegese0nned 9 ppamhageeneGs tavacea aN 60-0
3 too Leas. ie
: Oe: é NG ot | 7 J
° QO". ¥ ww doe B
4 x “~ :
Wet te Ri at
: POX", 0. o% VION a7
ne 3 QO” @- 25° of G os
tad os ov e° ‘ @ y,* os ve 6
= a4 i
aA Qo * tv" we 5
, H % 0 oe, : ;
as . -\9 “ *, in Se ie
: : “A te, ; i
Zio | ° “as. Mes a
: Ko) : a x '@) ; ry Ss a 2,
LZ: oy Y | | 4 " a te 1
4 dif. Long: , S33 sg es 00 ;
zoo) 6Lea: |
f cura

|
%

ies is
turn the neareft way unto L. Here many queftions may be -pro-

— —"

| not be the fame Longitude? —~

js eee ee oy ee PT le eR RT hE OS LT ere Sheer yy
i’ ee is ; .
e : ; 3

Sines aud Tangentsin Navigatim: == 193
pofed: | |
I. What is the Longitude of the Pore at O from L? ;

2. What isthe Longitude of P from N? And why O and P fhould

3. What isthe Rumb from O unto P ? , 7
4. Whatisthe diftance from O unto P? And why the way fhou'd

| be more from L unto P, going by O, then by N?

— — or - sove

5, What isthe Rumb from PuntoL 2

6, Whar is the diftance from P unto L ?

| 7. What is the Rumbd from N unto O ? ,

| ~8. What is the diftance from Nunto O? And why it fhould not be

the like Rumb and diftance from N unto O,as from P unto L?
& Thefe queftions well confidered, and either refolved by the Staff,or
| pricked down onthe Chart, and compared with the Globe and the
->commonSea-chart, fhall give {ome light tothe direction of a courfe,

and reduétion of places to their due Longitude, which are now fully

- deftorted inthe common Sea-charts,

Here follows all the ufual Problems of failing, according to Aferca-
tor, which are refolved Arithmetically by the Table of Logarithm
Tangents, without the Table of Meridional parts, and may alfobe
performed Geometrically, by the Fangent Line upon the Crofs-ftaft if
it be Jarge. )
_ Firft, we are to know that the Logarithm Tangents from 45 gr.oo m.
‘upwards, do increafe in the fame manner, that the Secants added toge-

ther do, if we aecount every half degree above 45 gr. oo m, tobe

‘one whole degree of CMercators Meridional Line ; and fo the Table
of Logarithm Tangents, is a Table of Meridional parts, to every

two minutes of the Meridian Line, leaving out che Radius in every

| The manner of making ufe of itthus, (as it is fhall more plainly ~
appear in the Examples of the following Problems ) becaufe the Ta-
bles begin at 45 gr. 00 m. and that every 30 m. isforawhole degree,
when one, or both Latitudes are’ given in any queftions, take the 3 of
leach Latitude, and add 45 gr. oom. to each of them, and take the
Tangent of the fum ef each, tor the equal parts of the Latitudes given
Cnegleétingthe Radius as beforefaid ) then fubtraét the leffer fum of
equal parts from che greater, and the remainer divide by the Tangent
ot 45, 30, the Radius negleéted, the Quotient fhall be the equal or
. Qa | Equinoétial

|

294 The vue nfe of Mescators Charts EC
Equinoétial degrees contained between the cwoLatitudes,or elfe mulei-
ply the forefaid remainer by 10,and divide it by the half ofthe forefaid
Tangent of 45 gr.30 m. and the Quotient fhall be the equal or Equi- —
noétial Leagues contained between the two Latitudes.

Example.

‘y Erone Laticnde be 45 gr. 30”. the 4 is 22 97. 45 on nto which »
Leas 45 gr. 00m. the {um is 67 gr. 45 m. the Tangent above rhe
Radius is 3 881591. ter che other Latitude be 40. gr. 00 m the $ is
20gr, OO m. Unto which add 45 gre OO”. the fum 1s65,gr, OO. the
Tangent above the Radius is 3313275 which fubtracted from the
former, the remainder is 56831: which being divided by 75803 the —
Tangent of 45gr. 307. above the Radius, the Quotient is7 gre 497 —
parts, the Equinoctial cegrecs contained between the cwo Lacitudes, —
or elfe multéply the remainder or difference 568316 by 1o and divide —
itby 37901, the 4 of the Tangent of 45 gr. 30m. above the Radius,
and the Quotient is 149 Lea. 94 parts, the equal or Equinogtial —
Leagues contained berween the two Latitudes, andthe like of any
other. 4

PROBL. I.

T he courfe and diftance that the Ship hath ven or (ailed, being given,

to find the trve place or point where the Ship % in Mercators

Charts | | if

- ,

“Daita Ship fail SS E 4 E128 Leagues from Latitude 4s gr. 30 me

f North Laticude, that 1s from A to EB, according «o the plain Sear

chart, I demand the crue place or point chat the Ship is at, accordingto

cMercators Chart. |
Before chis queftion can be refolved, we muft find what Latitude the

Ship isin, which is thus found : 7 uf

As the Radius is to the Sme of the Complement of the courfe ©

59 deg. 04m. SOI E A the diftance vpon thecourfe 128, to A Dehe

true difference of Latitudein Leaguss, which is 110, Thisbeing con

- werted intodeg. and min. is S$deg, 30%, and‘becaufe the Latitude de-

creafeth, or the Pole is depreffed, we fubtract it from 45d. 30 ™

and the remainder is 40 deg. 00m. the Latitude the Ship is in, thas 4

al

: The true ufe of MERALOIS Charés = 2S

| at E, according to the plain Sea-chart, or at C according to Mercator :
butbefore we can find the point C, we muft find the diftance of the
point B in the Meridian Line from A: the manner how to do it is fhew-
_ edinthe Example before this Problem, and it is there found to be 150
Leagues near. Now the point C, the true place of the Shipin Aercae

_ tors Chart may be found two feveral ways.

aay
| a
le ; rs
wis GN
Uy i ee
S: €
\ , ie o%
| RS x
| © =
| A i saat
| pat % oi 4
| DD iesensccrcnsncccesscocee sees sat EZ 4.0 = © ‘o}
a cilia oN ead
Bi. evecrssee onuecse ‘evegpyodargsevcouers G5 HODDOvaTEDESD Ke c Q=
“99 Lea: 40-00

Firft, As D A the true diftance of Latitude 110, isto AE the true
_ diftance run upon the courfe, fo is BA the difference of Latitude en-

larged 150, to AC 174}, the enlarged diftance, which being laid off
~ ppon the Line of che courfe, gives the point C, the true place of the
_ Ship in Mercators Chart.
[ Qq 2, Here

t. Pe re ks ya ee
5 - CATs
+

296 ee nr ale of Mercators Chart.

Here we may take notice, that the rue point of any place, according —
to the plain Sea-chart, or according to Mdercators Chart, is always upon

one and the fame riphe Line of the:courfe.

‘Secondly, As the Radius is to che Tangent of the Courfe 30 4.56 m.

fo is A B the difference of Latitude enlarged 130,to BC go the diffe-
rence of Longitude,which being laid off upon the Perpendicular BC,

gives the point C, the true place of the Ship in AZercators Chart.
PROBL, IL I
The cosr{e that the Ship hath failed on, and both Latitudes being known,

to find the true place or point that the Ship w on in Mercators Chart,

and the true diftance that the Ship hath failed.

q\Uppofe a Ship to fail SSE% E from che Latitude of 45 deg. 30m. :

‘until it be inthe Laticude of 40 deg. 00 m. that is from Ato E, ace
cording to the plain Sea-chart,or from Ato C,according to Mercators
‘Chart.

Firft, We muft find the difference of Latitude enlarged, as is be-
fore direGed, which is A B 150 Leagues,

a, Asche Radius is tothe Tangent of the Courfe, 30 deg.:56 m. fo ,
is AB 150, toBC ogo, which laid off upon the Perpendicular BC, ©
gives C che true place or point, which the Ship is on intMercators

‘Chare, | ¥

2, Asthe Sine Complement of the Courfe 59 deg. o4 m. is tothe, -
Aadius, fois D Athe true difference of Latitude 110,to AE the true —

diftance run upon the Courfe 128.

PROBL. IIL.

Both Latitudes given, and the diftanceran upon the Courfe, to find the |

~

point or place that the Ship 13 on in Mercators Chart, and the conrfe Al

or point of the Compas that the Ship bath failed on.
sul ie a’Ship-to fail 138 Leagues, between South and. Eaft,from A

inthe Latitude of 45 deg. 30 #2. and at the end of her diftance, it

ibe inthe Laticude of 40 deg. com.
Firft, Find the difference.of Latitude enlarged,asis before directecs
which is AB 150, ;
; LAs

| The true ufe of Mercators Chart. 297
__ 1.AsDAT1io, the true difference of Latitude, is to AE 128the
| true diftancérun, fois B A1so thedifference of Latitude enlarged,
toAC 174 2; the diftance enlarged, which laidoff upon the Line AE,
_ from A to C it fhall be the true point or place that the Ship is onin
| Mercators Proje&ion. - ert oe

_ 2, AsAE 128, che true diftance run, isto A Dio, the true diffe-
-renceof Latitude, fo is the Radius to the Sine of the Complement of -
the courfe 59 deg. 047. which Complement 59 deg. 04 ms. fubtra&
outof go deg, 00 m, and the remainder is 30 deg.056 m.the courfe,and

| being itis becween South and Eaft, itisSS E 3 Bafterly.

es PROBL. IV.

| Both Latitudes, and the departure or diftance of the Meridian yo4 are
| — apon, and the Meridian you began your conr{e on, to find the point or
| place where you are in Mercators Chart, alfe the courfe that you
| have made good, and the diftance that you have run frou the place,
where you began your cour fee ,

| His Problem is chiefly ufeful for the Navigator, when he hath caft -
_ HL up histraverfe. AdmitaShipto{ail upon the Southeaft quarter of
the Compafs, from Latitude 45 deg, 3.0 ms; unto Latitude 40 deg. 00 m,
and the departure from the Meridian ic went from, be 65 2, Leagues.
| Firft, Find the difference of Latitude enlarged, as is before directed
150 Leagues. |
4. AsAD rrothetrue difference of Latitude, isto DE 65 2,the
departure from che Meridian, fo is A B 150, the difterence of Lati-
‘ude enlarged, to BC 90 Leagues, the diference of Long tude,which -
laid off upon the Perpendicular B C, from B to€ fhall be the point or
—dlacein Adercators Chart, where the Ship is.
_ 2, As AD irothetruedifterence of Latitude, isto DE 65 £,the
Jeparture from che Meridian, fo is the Radius to the Tangent of che
sourfe 30 4, 5 6 m. thatis two points 3 from the Southto the Eaftward,
hatisSS E4% E the courfe that the Ship hath kepr. |
_ 3. Asthe Sine of the courfe 30deg. 56 m. is tothe Radius, fo is
DE 6s +, the departure from the Meridian, co£ A 128 the diftance
un,

PROBL:

i oa a

298 «= sis he tre wfeof Met@ators Charte

PROBL. V.

Buth Latitudes being given, and the difference of Longitude, to find the
diftance the Ship hath kept, and the diftance it hath ran. a
Dmit a Ship to be at A in North Latitude 45 deg. 30 m, and to fail
~ MSoutheaftwards, untill ic be at Ein Latitude 40 deg. OO m. accord-
ing to the plain Chart, and the point C be the place in AZercators
Chart where the Ship is, and the difference of Longitude be B C90
Leagues. |
: Fictt, Find the difference of Latitude enlarged, as is before directed
150 Leagues. }
“1, AsAB rgothe difference of Latitude enlarged, is to BC 90,0
is the Radiusto the Tangent of che courfe, 30deg, 56. which istwo »
points thatisSSEZE. |
2. Asthe Sine Complement of che courfe 59 deg.o4 m. istothe Ra- |
dius, fo is DA 110 the true difference of Latitude, to A Ethetrue :
diftance run128. }

}

PROBL, VI.

‘One Latitude, with the courfe, and the difference of Longitude given, to
find the other Latitudes and the diftance run. o

gn Uppofe aShip to be inthe Latitude of 45 deg. 30%, North Latia
Sak andto failSSE4E ( until the difference of Longitude be:
-90 Leagues ) that is from Ato C, which is the pointor place of the

Ship in AMdercators Chart. Bt

1. Asthe Radius is tothe Tangent Complement of the courfe 59 4.
‘04m. fo isC B the difference of Longitude 90, to AB1so the diffe-
rence of the Latitude enlarged, by which multiply 37901 the ¢ of che
Tangent of 45 deg. 30 m above the Radius, and divide theProdu&
by 10, and the Quotientiss68515. Thentake the Latitude given,

the 3 of 45 deg. 30m. which is 22 deg. 45 m. unto which add 45 deg,

‘oo wm, the fum is 67 deg. 4.5m. then feek the Tangent of 67 deg. 45 mm
above the Radius, which is 3881591, and fubtra@& the former Quo-

tient 568515 from it, and the remainer is 3313076, which feek inthe

Tangent, and you fhall find it at 65 deg. oom. from which fubtract
i 45 eg.

yy

The defeription of the Bow. ap.

"45 deg, 00 m, the remainer is 20 deg. 00 m. which being doubled, is

4.0 deg, oo m. the Latitude required. Here we are to note, that if che
Latitude had increafed, we muft have added the Quotient 568515 to
the Tangent of 67 deg. 45 m. and fo fought the fum in the Tangents,
to have found the Latitude required. .

2. Asthe Sine of the Complement of the courfe 59 deg. 04m. Isto
the Radins, fo is D A che crue difference of Latitude 110, to-A E, the
true diflance run 128. age
Although Ihave fet down but the proportions and the anfwers to

each gueftion, they may all be calculated by the Canon, and the Chi- |

liad of Logarithms in this Book, - 7

APPENDIX,

Concerning the defcription and ufe of an Inftrument, made in forms:
of a Crofs-bow, for the more eafie finding of the Latitude at Sea.

BT ie former Prop. fuppofe the Laticnde to be known, I will here
fhew it how ic may be ealily obferved. |
Uponthe Center A, and Semidiameter AB, defcribe an Ark of a: °

S © Tee Keay thy

ria ladle The defeription of the Bows :

‘ f .
6 3 a ¢ at

CircleS BN. The fame Semidiameter will fet off 60 gr. fromB unto$ ‘
for the South ead,and other 60 gr. from B unto N for the North end
of the Bow: fo the whole Bow will contain 120 gr. the chird part of a

Circle. Let ic therefore be divided into fo many degrees, and each
degree fubdivided into fix parts,that each part may be ten minutes: bug
lee the numbers fet to itbe 5,10, 15, unto 90gr. and then again 5,10,

TS, unto 25, thats 5 may fallin che middle, as in this Figure. :

The Bow being thus divided and numbred, you may fee the months
and days of each month upon the back, and fuch ftars as are fic for ob-.
fervation upon the {ide of the Bow. |

If you defire to make ufe of it in North Latitude, you may number
23 £r. 30 m, from 90 towards the end of the Bow at N, and there place
the centh day of Pane, And 23 gr. 30m. from ge towards S; and
there at 66gr. 30 ms. place the tenth day of December. And tothe
reft of the days of the year, according tothe declination of the Sun
‘atthe fame days.

The {tars may be placed in like manner according to their Declina-
tions, to the year 1670, ;

Ar&urus 20 gr, 57 me | :

The Bulls Eye Ig +47

The Lions Heart IDs alg2

The Vultures Heart 8 8 Mm

The little Dog 6 o from go toward: the -
North end of the Bow at N. Then for Southern ftars, you may number —

their declination from 90 toward the South end of the Bow atS. As |
firft che chree Scars in Orions Girdle,

te Orta RF irht at O gr. 37 m. ‘|
Girdle the, Pica eee 7
Third 2

?
The Hydra’shearte 9 16

The Virgins Spike 9 24

The great Dog 16 9

Aquaries Leg PF 0.633

The Whales Tail 19. 48

The Scorpions Heart’ 25 37 |

Fomahant 30 16 And fo che South crown,
the Triangle, the Clouds, the Crofiers, or what other Stars you think |

fi for che obfervatior, This I call the forefide of the Bow. 4

3 If

|

- —— es
as Pe

eS t
b; ae

Fi seriptinef a bed Bows

| Mee If you defire ta Hake ufe of i¢in South Latitude, you may turn the
‘Bow, and divide the back fide of it, and number it in like manner,
and then put on the months and days of the year, placing the tenthof —
“December at the Sourh end,and thetenth of Jase coward the middle of —
hic Bow, and the reft of the days according co the Suns declination as —
efore. ,

ioe iF

jor

The chiefeft of the Northera Stars may here be placed ia like

The Pole Star at

The firft Guard

The fecond Guard
The great Bears back
In the great Pe Ai d
Bears Tail Third
The fide of Perfeus
The Goat
The Tail of the Swan
The Head of Medufa
The Harp

Caftor

Pollux

The North-Crown
The Rams Head’
Arcturus

The Bulls Eye

The Lions Heart
The Vultures-Heart
Orions right Shoulder
Orions left Shoulder

_ manner, according to their declination, Asno 1670.

2 32m,

34

| And fo sny other Star whofe declination is ; knowa unto yOu 5
. wit being done. Theufe of this Bow may be,

i 1. The day of the month being known,to find the declination of the San,
| 2. The declination being given, to find the day of the month.

_Thefe two per depend onthe making of the Bow. If theday be ©
Re known,

ek eas a eg

DA -
oe ga

oe ata
age ore poe ei Z

AE

be

302 . pheufe of the Bow

+. Pe oe

known, look irout in the back of the Bow: fo the declination will ap-_
pear inthe fide. Or if the declination be known, the day ofthe month _
is fer over againft it. As if the day of the month wére the 14 of July,
Jock for this day in the back of the Bow, and you fhall find it over —
apainft 20 gr. of North declination. If che declination given be 20gr.
tothe Southward, you fhall find the day to be either the eleventh of

Nevember or the eleventh of January. ) ae

3. To find the Altitude of the Sun, or Stars.

Here it is fit to have two running fights which may be eafly moved —
onthe back of the Bow. The upper fight may be fet either to 60 gr. OF
t0 70 gr.or to 86 gr. as you fhall find to be moft convenient: the other”
fight may be fet on to any place between the middle andthe other end ©

of the Bow. Thenwith the one hand hold che Center of che Bow ta:

your eye, fo as you may fee che Sun or Star by the upright fight, and
with the other hand move the lower fight up or down until you have —
brought ene of the ecges of it, to beeven with the Horizon Cas when —
you obferve with the Crofs-ftaft) fo the degrees contained between —
chat edge and the upper fight, fhall fhew the Altitude required. ‘a

Thus ifthe upper tight fhali be at 80 gr.and the lower light at 50 gre |
the Alcitude required is 30 gr. . J ai

4. To find any North Latitude, by the Meridian Altitude of the Sun ata
“forward obfervation,knowing either the day of the month, or the decll.
nation of the San, a

ies

‘As oftas you are to obferve in North Latitude, place both the fights
‘on the forefide of the Bow, the upper fight atthe Declination of che |
Sun, or the day of the month at the North end, and lower fight coward —
the South end. Then when the San comethto the Meridian, turn your
face to the South, and with the one hand hold che Center of the Bow co
your Eye, fo as you may fee the Sun by the upper fight; with the other
hand move the lower fight, until you have brought one of the edges of |
it to be even with the Horizon: fothatedge of the lower fight thall
thew the Laritude of the place in the fore-fide of the Bow. a |
Thus being in North Latitude upon the ninth of October : if I fet
the upper fight to this day, at the fore-fide and North end of the Bow, |
1 thall find it to fall to the Southward of g@ upon 8ogr. and therefore |
rs

—

st

|
|
|

day of September, the formoft guard at 75 gr. 45 m. the hindmoft gu

at 73 gr. 25 m.and the reit ac
tioned, fo the go deg. fhall reprefent the North Pole of the ae

50 gr.
| 50gr.h

the Northward may be placed either on the backfi

2k ae ia me ce Setic te OC ‘G
Fay a Soe ed ta So Lat ae * roe eee Pal sa
Bs + yy a Ne ath Sad aoa

| ig finding the Latitades oo gas

E at 10 gr. of. South declination. Then the Sun coming to the Meridian, [
may ier the Center of the Bow to mire eye, as if I went to find che Al-

tirude of the Sun, holdingthe North end of che Bow upward,with the

upper fight between mine eye and the Sun, and moving the lower fight,
until it come to be even with the horizon. Ifhere the fower fight fhall

ftay at 50 gr.. I may wellfay, that the Latitude is 90 gr. For the Me-

| ridian Aleicude of theSunis 30 gr. by the third Prop. and the Sun ha-_

ving 10 gr. of South declination, the Meridian Altitude of the Equa-

- tor would be 40 gr. and therefore the obfervation was made in 50 gr. of

North Latitude.
| By the fame reafon, if the lower fide had ftayed at 51gv. 90 ms. the

Latitude muft have been 51 gre 30 m. and fo inthe reft.

5. Te fiod any North Latitude, by the Meridian Altitude of the Stars to
the Sonthward.

fetto the Star, which you intend to obferve,
deof the Bow. Then hold the North end of
the Bow upward, and turning your faceto the South, obferve the Me-
ridian Altitude as before: fothe lower fighs hall thew the Latitude
of the place in the fore- fide of the Bows —

Thusif in obferving the Meridian Altitude of the great Dog -ftar,
the lower fight fhallftay at 5927. it would thew the.Latitude to be
For this Star being here placed at 73 gr. 48 m. if wetake thence
‘s Meridian Altitude would be 23 gr. 48 ws. to this if we add
e South declination or this Star, it would thew the

Let the upper fight be
here placed in the fore-fi

10 or. 12m. forth

Meridian Altitude of the Equater to be 40 gr. and therefore the Lati-
tude to be 50 gr.

6. Tofind any North Latitade, by the Meridian Altitude of the Stars to
the Northward. .

If the Bow be intended only for North Latitude it may fuffice to

have the degrees divided only onthe fore-fide, and then the Stars to
deor the infide of

the Bow by thefe degrees: the Pole-flar at 87 gr. 20m. near che 20.
ard

cording to their declinations before mess
Rr 2 hen

.

y

304 i eke The ufe of the Bow - NOS Sa

_ of the Star when he cometh to be in the Meridian, and under the Poles -

— of che Bow,

t an,

we

“When any of thefe ftars come to be inthe Meridian, and under th “

Pole, fectheupper fight torhat Star, hold the North end of. the Bow.

upward, and turning your fsceto the North, obferve his Altitude as
before; forthe degrees contained between the 90 degrees and the lower py
fight, thal thew rhe Alricude of the Pole. Lat a
Thus the former guard coming to be inthe Meridian,under the Pole,
if you obferve and find the lower fight ro ftay at 40 or, the Elevation —
of the Pole is 50 gr. according tothe diftance between 40 andgo, s
If you would obferve any of thefe Scars, ac fuch time as they come —
to be inthe Meridian, and above the Pele, you may place chefe Stars

‘in the Bow above go gr. the North Scar at 2 gr..40 m. near the fourta —

day of September,the formok Guard at 14.97.15 m.the bindmoft Guard, |
at 16 gr. 3§ m. and fuch others as you think firreft,according to their —
diftance from the Pole: then fetting the upper fight to che place of
Star above the Pole, the reft of the obfervation willbe the fame as __
before. aie , | ry
But if the Bow be made to ferve at large, both in South and North \
Latitude, then thefe Northern Stars.would be let placed onthe back. ~
fide of the Bow, by the degrees onthat fide, according tothe Com-
plement of their declisations, thatthe North Stars may anfwer‘to thea
North Sun in South Latitude, ia fuch fore asthe Southern Stars didta.
the South Sun in North Latitude in the former Prop. This being done,
lecshe upper light be fet to the Star which you intend to obferve, here
placed on the backfide of che Bow. Then hold tae North end of chely))
Bow upward, and turning your face to the North, obferve tke Alticude ;

fothe lower fight fhall thew the Altitude of the Pole inthe backlide

im, |
Thus the former guard coming to be in the Meridian underthe © |

i |

Pole, if you obferve and find.the lower. fight to ftay at.9q gr fuchis
the Elevation of the Pole, and the Laticude of the place to the North-_

ward. For the diftance beeween the two fights will fhewthe Altitude _
tobe 35 gr.45 mm, and the Star is 14 gr. 15 m. diftane from the North —
Role. Tneferwodo makeup sogr. for the Elevation of the North.
Pole, and therefore fuch is the North Laticnie. ie |

|
|

a f |

in finding the Latitude, | e : "

9. To find any Sonth Latitude, by the Meridian Altitude of the Sun at a
forward obfervation, knowing either the. day of the month, or the
declination of the Sun. . :

| When you are come into South Latitude, turn both your fights to the
| back fide of the Bow: the upper fight to the declination of the Sunjor
the day of the month at che South end, and the lower fight coward the —
| North end of the Bow. Then theSun coming to the Meridian, turn
- your face tothe North, and holding the Southend of the Bow upward,
obferve the Meridian A'titude as before : fothe lower fight fhall-fhew
| the Latitude of the place in the backlide of the Bow.

| Thus being in Souelr Latitude, upon the tenth of 4227, if you obferve |
and find the lower fight co {tay at 30 gr. on the backfide ‘of the Bow,
| fachisthe Latitude. For the declination is 20 gr. Northward, the Al-

| titnde of che Sun between the two fights 40gr. the Altitude of the

| Equator 60 gr. and therefore the Latitude 30 gr. 3

l i 8.70 find any South Latitude by the Meridian Altitude of the Starste -
|. the Northward. ;

| Lettheupper fight be fet tothe Scar which yau intend to obferve,
here placed on the backfide of the Bow. Then hold the South end of
the Bow upward, and turning your face tothe North, obferve the Me»
| ridian Altitude as before : fo che lower fight thall fhew che Laticude of
\the place inthe back fide of the Bow. |

Thus being in South Latitude, and the former guard coming to be in
the Meridian overthe Pole. If you obferve, and find che lower light to
| ftay et 5 gr. fuch isthe Latitude. For the Star is I4.gr, 15 m. from the
North Pole, the Altitude of the Star between the two fights ogr.t5 m.
the North Pole deprefled § gr. and therefore the Latitudes grtothe .
—Southward.

| 9, Toobferve the Altitude of the Sun by the Bew,or with an Aftrolabe. -

_ Here itis fit to have athird fighe ( like co the Horizontal fight be- -
fonging tothe ftaff ) which may be fet to the Cenrer of the Bow.
If the Sun be near to the Zenith, bold the Bow as when you obferve
with the Affrolabe, fo asthe Center being downward the Line AB .
a | may. ,

506 ay The wife of the Bove Pee

may be vertical, and the Line S N Parallel co ye Horizon, rhe funn: ‘
ing one end of the Bow toward the Sun, you may move one of the
‘fights on the back of the Bow, until the fhadow thereof fall on the —
- middle of the Horizontal fight, fo the degrees contained between the
Vertical A B, and that upper fight fhall thew the diftance of the Sun
fromthe Zenith.

If the Sun be nearer to the Horizon, you may hold he Bow fo as

the Line SN may be Vertical, and the Line A B Parallel co the Hori-
zon, then obferving, as before, the degrees contained between the

Line AB, andthe ppper fight, hall fhew the Alshude of the Sun
-above the Horizon. ‘

10, To find a South Latitude by the Meridian Altitude of the Stars tothe i:
Sonthward.

Lec the upper fight be fet to the Star which you intend to obferve,
which might be here placed on the foots fide of the Bow by the Coma :
plement of their declinations, if we knew the true place of fuchas |
are near _to the South Pole.

Then hold the Southend of the Bow upward, and turning your face. .
co the South, obferve the Altitude when he cometh to bein 1the Meri-
dian, and under the Role, fo the lower fight hall fhew the Altitude of |
the Poleinthe forefide of the Bow. |

|
It. To obfervethe Altitude of the Sun backward. a

Set the upper fight either to 60, or 70, Or 80 gr. as you thal! find
to be moft convenient, the lower light on any place between the mid-
dle and the other end of the Bow, asd have an Horfz yntai fighe to be
fer tothe Center. Then may you turn your back to the Sun, “and chen
back of the Bow toward your felf, lookirig by the lower fizht through |
_ the Hor‘zontal fight, and moving the lower Gght up and i down until: |

ghe upper fight do cafta fhadow upon the middle of tke Hor-zontal)
fights: fothe degree, contained between the two fights oa the Bom
thal! give che Aleicude required. |

Thus ifthe upper Gghe fhall be at $0 gv.and the lower fight at « sogre |
the Alcitude required 1s 3 Ogr.as inthe tl third Prop.

QO: if you eurnche other end af the Bow amends and fet the upper’

fight to the beginning of the Quadrant, and then obferve as belt
she lower fight will fhew the Alucudce

12

+ 12670 find any North Latitude by the Meridian Altitude of the Sup at
| aback obfervation, knowing etther the day of the month, or ehe de-
clination of the Sus. ;

6
:
ie
| 8

= Sar bs

Place your three fights as before onthe fore-fide of the Bow: the
upper fight to the declination of the Sun, orto the day of the month,
at the North end ; the lower fight toward the South end of the Bow;

and the Horizontal fight to the Center. Thenthe Sun coming tothe |

Meridian, turn your face tothe North, and holding the North end
of the Bow upward, the South end downwards, with the back of it

“toward your felf, obferve the thadow of the upper fightas in the for-

mer part of the fifth Propofition, fo the lower fight fhall fhew the Lati-
tude of the place inthe torefide of the Bow.

Thus being in North Latieude upon the ninth of Odfober, if you:
obferve and find the lower fight to ftay at so gr. on the forefide of

| the Bow, fuch is the Latitude. For the declination is 19 gr. South-.
ward, and the Altitude of the Sun between the two fights 30 gr. the .
| Altitude of the Equator 40 gr. and therefore the Laticude 50 gr.as in.

the fixch Prop.
13.70 find any South Latitude by the Meridian Altitude of the Sun at
aback obfervation, knowing either the day of the month, or the de-

clination of the Sun.

When you obferve in South Latitude, place your three fights on.

| the back fdecf the Bow: theupper fight tothe declination of the
“Sun, or the day of the month atthe Southend; the lower fightto-
-wardthe North endof the Bow, andthe Horizontal fight tothe Cen-
ter. Then the Sun coming tothe Meridian, turn your face to the.
South, and holding the South end of the Bow upward, with the back

of it toward your felf, obferve the fhadow of the upper fight as be-.

- fore: fo the lower fight fhall thew the Latitude of the place in the

| back fide of the Bow. .

een ws =: eee sees

Thus being in the South Latitude upon the tenthof Afay, if you
obferve and find the lower fight to ftay at 30 gr. onthe back of the.
Bow, fuch is the Altitude of the Sun between the two fights 40 gr. the:
Altitude of che Equator 60 gr, and therefore the Latitude 307. asin
the feventh Prop. : 14,70,

“a finding the Altitude, — ty.

ey le aE

Place your three fights according to your Latitude: the Horizontal ‘i
- fight cothe Center, che lower fight tothe Letitude, and upper fight
- among the months. -Then when the Stin cometh tothe Meridian, ob-

_ oblerving find the Altitude of the Sun between that and the upper

_ then if you find che upper fight uponthe tenth of O&ober, and the eles
— venth of February, the queftion will be fcon refolved. _

_ lower fight, and the Scar by the upper fight, thea will the upper fight i"

_ >For by this obfervation che diftance of this S-er from the South Pole

RB re Va ree, 7. La J .

ge0 ieee she Bo

14.76 find the day of the monthyby kaowing the Latitude of the place, and =
obferving the Meridian Altitude of theSun, =

ferve the Altitude, looking by the lower fight through the Horizon- —
tal, and keeping the lower fight ftill at the Latitude, but moving the —
upper fight until i: give fhadow upon the middle of che Horizontal —
fight: fotheupper fight fhall fhewthe day of the month required. "7

Thus in our Latitude if you fee the lower fighe to 51 gr. 30m. and

fight to be 28 gre 30m. this upper fighe will fall upon the ninth of a
Oéteber, and thetwelfth of February. And if yet you donbe which of aa
them two is the day, you may expeét another Meridian Alticude 5 and

15 To find the declination of any unknown Star, and fo to place it on the
Bow, knowing the Latitude of the place, and obferving the Meridian
ltisnde of the Star. ee
When you find a Star in the Meridian chat is fic for obfervation, Ser
che Center of the Bow to your eye,the lower fight tothe Latitude,and

move the upper fight up or downunt'l you fee the Horizon by the |

{tay at the declination and place of the Star. ty
Thus being in 20 gr. of North Laticude,af you obferve and findthe
‘Meridian Altitude of the head of the Cofier to be I4gr.5Om., The —
upper fight will ftay at 34 gr. 50 m. and there may you place this Star. —

fhould be 34 gr. 50 mand the declination fromthe Equator 53 £71 mae
And fo for the reft, A |

The Stars which I mentioned before, do come to the Meridian 1a)

this order after che firft point of Aries

° neha

16.To find any North Latitude on land by obfervation with Thread and |
Plammet. tie Et

BBs
4,

Set the fight to the day of the month at the forefide and South end _

in fading the Latitude. ae $99

of the Bow; then when the Sun cometh to the Meridian curning the
North end in your left hand toward the South, foas the faght at che

Center may fhadow the fight at the day, obferve where thechread fal-

Jeth, and abate 20 gr, if icfallon 70 gr. the Laticude is sogr. If on

|

77 gr. 30 m. in the Laticude is 51 gre 30m. And fo in the reft,
Ik che Bow had been made only for finding che Latitude on Land I

_ mighe then have fet fuch numbers so it as needed no allowance.

|

17.70 fiad any Sonth Latitude on Land, by obfervation With Thread
and Plummet. |

Setthe fight to the day of the month, atthe backfide and North end

| ofthe Bow, and when the Sun cometh tothe Meridian, turning the
| Soith end to your left hand coward the North, obferve asbefore, and

abate 20 degrees.

| Or you may fer the fight to the day of the month, at the forefide,

and North end of the Bow, and fo obferving as before, the Thread will
fall onthe Complement of the Latitude.

The right Afcenfion of thefeStars‘is tothe year 1670.

| Hi. M. A, Mf,
The Pole Star at Oo 31 | TheLions Heart 9 5°
The Rams Head 1 48 | ThegreatBearsBack 10 43
| The Head of Medufa 2 471 Firftingr. BearsTail 12 39
_ The Side of Perkus 3 00 | The Virgins Spike Ey
_ The Bulls Eye 4 17 | Secondingr.BearsTail 13 12
| The Goat 4 §2 | Thirdin gr. BearsTail 13 36

| Orions Jefe fhoulder 5 07 | Aréturus i4, 02
POrions the firft 4 § 1§ | The formoft Guard 14 §3
| Girdle che fecond 5 19 | The North Crown Ig 21
ibe the chird 3 24 | The hindmoft Guard Ig 27
Orions right fhoulder 5 37 | Scorpions heart 16 Io

The great Dog 6 31 | TheHarp e425.
Caftor 7 13 | Vuleurs heart 19 35
The little Dog 7 22 | Swans tail DOSER Ce ar $s
Pollux 7 2§ | Fomahane | 22 38

oy | |

_ The Hydra’s Heart

Sf [ einne

Fir Guard
i>econd Guerd

Right Foor ; |

ee ee ee ee

{n the fquare :
L

j{n the mid- |
41

Le Siee es Tent 4 “:
Auriga os eros We ete:
(Head | 234 | sg 15 |
iL eft thoulder_ i 373 OF LAS 36
i dircus ¢ ZA ag
|Righe fhoulder | Shot ih ELE |
ES eter emeren °
Cepheus Gir- — }.
idle eh Sie 69 vie
Right thoulder » 316 10:'60 - 24
“eft fhoulcer | 340 16164 33 :
dead 329 «644i 56 a3 y
jRightfoot | 394 551176 252
Left foot Tie iee See ab te.

dle of the ArftS

In the firft 2

ip the fecond

| ‘the third
winding 2
{After the

; i 58

"Fhe End of the ee Book:
of the Crofs-Staff.

t

\T, he right Afcenfian, ‘Declination, and v

Magnitude of fome principal Fixed Stars

See CESSES EA SSE a ORR CLL (RAE CERRITOS el ewougeraesy fitness

The Stars Names,

The Pole-Star

The Girdle of Andromeda

The former Horn of the Ram
Bright Star in the Ram’s Head
_ The Whale’s Jaw

The Head of Afedufa

The Bull’s Eye

} The Goat

The former Shoulder of Orion
The latter Shoulder of Orion
The great Dog

The uppermoft Head of the Twins
The little Dog

The lower head of the Twins
The Crib

Hydra’ s Heart

Lion’s Heart

Lion’s Loins

Lion’s Tail

The Virgin’s ‘pig

Aliot

Vindemiatrix

The Virgin's Spike

Arhurus .

The Southern Balance

The Northern Balance
Inthe Serpent’ s Neck
The.Scorpion’s Heart
Hercules Head

Opbinchus Head

The Harp

The Vulture -°

The upper Horn of the Geat
Left Hand of Aquarius

Left Shoulder ot Aquarius

| Pegafus Mouth

Right Shoulder of, Aqusrte
Fomahant

SS CS ee GE) SED

Right Decli-
pices nation.
it 7 87 i
12 pee 33 48
eae
26 gG} 21 48
, 4% | 2 ae
4l 27139 35
64 Oo | 46
7% 44] 45 a
| 76 38 | 4 39
Bigo eae i"
97 27416 13
| co | | 3214.35
ve Sid a 6 6
rT. o 4 38 id
125 444200452
1397-39217 |
147 4714133 39
163 $4] 22 26
17z 49 Ee 32
EE 321 § |
139 36 ie 36 |
19t Ap | i2z §1
196 441 9 17
21209 sé] 2 4
| rig *§6 1.14 *|
<8 2m 304 Br
‘P23! 49). 7 35
| 242 | 25 34 |
$254 427,74 §1
(259 4 F258
| 76 17 | 38 3
S293 ~ 2719) Bid
Fe 2 | 4
307 Io IQ 43
318 18 q 2
321 49} 8 18
|e 59 | I 58 |
339 t 23
341 3 «I |
3 15

PLLLLZLY LLLZOZLZ

ZZLZLZLZLZ

SS

. mee Se ee eee Gals, SS

Magni-

tude,

eee

2
2
4
3
z
3
1
I
2
I
z
z
2
N
2
1
2

~

Bie Ae.

THIRD BOOK

i

The Ufe of the Lines of Numbers,Sines
- -and Tangents, forthe drawing of Hour-lines on i
all forts of Planes. | i
Here are cen feveral forts of Planes, which take theirdenominati= = —-
on from thofe Great Circles co which they are Parallels, and ~

Ds may fufficiently for-our ufe be reprefented in this one Fanda-
i mental Diagram, and be known by their Horizontal and Perpendicular ~
i a of fuch as know the Latitude of the Place, andthe Circlesof the
| Sphere. 3 aa
ij, 1. An Horizontal Plane parallel co the Horizon, here reprefenced by the
: outward Circle ES W No | LP ey

2. A Vertical Plane, parallel torhe prime Vertical Circle, which paf-
| feth through the Zenith, and the Points of Eaft and Weft in the Hori-
zon, and is right to the Horizon and the Meridian 5 that is, maketh righe
| Angles withthem both, Thisisreprefented by EZ W.

aah nagy Oak

3. A Polar Plane parallel co the Circle of the Hour of 6, which pafleth
| through ‘the Pole, and che Points of Eaft and Weft, being right to the
~ Equinodial and che Meridian, but inclining to che Horizon, with an Angle
‘equal tothe Latitude. Thisis here reprefented by E P W.
4. An Equinogual Plane parallel co che Equinoétial, which paffeti
_ through che Points of Eaft and Weft, being right tothe Meridian, but in-
| clining co che Horizon, with an Angle equal to the Complement of the
| Latitude. This is here reprefented by E A W. |
> 5. A Vertical Plane inclining -to the Horizon, parallel to any Great
| Circle; which paflech chrough che Points of Eaft and Weft, being right
to the Meridian, bac inclining tothe Horizon, and yer not paffing through.
the Pole, nor parallel to the Equinottial, This isherereprefented either by ~ oe
EILW, or EY W,orELW. . ae

Aaa 6A

a

a The Diftinition of Planes; | ii
: i: 3,

IN

6. A Meridian Plane parrltel to the Meridian, the Circle of the Hour
of 12, which paffech chrough the Zenith, the Pole, and Points of the
South and North, being right to the Horizon, and the prime Verticals’
This is here repretented by § Z'Nz. : 7 t
7. A Meridian Plane inclining to the Horizon, parallel ‘toany Great
Circle, which paffech through che Points of Souch-and North, being right
' ~- to the prime Vertical, but incining-to the Horizom, This is here reprefente —
ed by.S GiNe. . oa
3. A Vercical Declining Plane, parallel to any Great Circle, which paf=
‘eth through the Zenich, being right to the Horizon, but wclining co che
. . Meridian, This isreprefenced by BZ Dy A |
+ Eat

~

|

1 a
Wy
i

|
|
|

!

“A Deferlpiton of the Fuindundital Disgraws
9. A Polar Declining Plane, parallel to any Great Circle, which paffec
through the Pole, being right tothe Equinodtial, but inclining to the Me

tidian. This is here reprefented by H PQ. | !
_ to, A Declining Inclining Plane, parallel to any Great Circle, which

as righeto none of the former Circles, but declining from the prime Ver

tical, and inclining both tothe Horizon and the Meridian, and all che

_ -Hour-circles. This may here be reprefented either by BM D,or BE Dyot

BK D, orany fuch Great Circle, which paffech neither through the South

and North, nor Eaft and Welt points, nor throvigh the Zenich, nor che
[BPole:. eg A Ta oi

Each of thefe Planes (except the Horizontal.) hath ewo Faces whereon

Hour-lines may be drawn, and fo there are nineteen Planes in all. The

Meridian Plane hath one Face to the Eaft, and another to the Weft: The —

other Vertical Planes have one co the South, and another to the North,

|
|
|

_ and the reft one to the Zenich, and another to the Nadir: but whatis faid
_ of the one, may be underftood of the other.

Todeforibe the Fandamental Dicgrams:

_. The Defcription of this Diagram is fet down at large in the Ufe of my
_ Settor, Chap, 3. But for this purpofe it may fuffice, if ichave che Vertical
| Circle, the Hour-circles, che Equator, and che Tropicks firft drawn in it, -
_ other Circles may be fupplicd afterwards, as we thall have ufe of them zg
| And thofe may be readily drawn in this manner.

Let the outward Circle reprefenting the Horizon be drawn,and. divided

“into four equal. parts with § N the Meridian and EW the Vertical, and
each fourth part into 90 gr. That done, laya Ruler to the point S, and

each Degree in the Quadrant EN, and note the Interfe@tions where the

Ruler croffeth che Vertical, fo thall the Semidiameter E C be divided into
‘other 90 gr. and from thence the other Semidiameters may be divided in
‘the fame fort. Thefe may be numbred with 10, 20, 30, &c. from E toa
ward C, and for variety with 10, 20, 30, &s. from C toward W. But
forthe Meridian, the South part would be beft numbred according to the

Declination from che Equator, and the North part according tothe di«

-fance from the Po'e.

_ Then with re{pee unto the Latitude,which here we fuppofe tobe 54 gr.

30m. open the Compafies unto 38 gr. 30m. from Coward W, an
_ prick them down in the, Meridian from C unto P, {o this point P fhalf
it

eprefenc thePole of the World, and through it maft be drawn all the
our-circles, 7 Aaaz Having

~ South pare of the Meridian at 75 gr. from the Zenith, and t

eee a aa os ame Rad UALR yo oa SY Ee ene Ores Bh:
& _- Defeription of the Fundamental Diagram. |

Meridian a licele without the point S$; and draw chem intoa Circle E PW,,
which will be the Circle of the Hour of 6% | ) I
Through this Center of the Hour of 6, draw an occult Line at lenoth-
parallel to EW, fo this Line hall contain the Centers of all the other
Hour-circles,. Where the Circle of the Hour of 6 croffeth this occult
Line, there will be the Centers of the Hour-circles of g and 3: The dix
ftance between chefe Centers of 9 and 3, will be equal to the Semidiame.
ters of the Hour-circles of no and 2: and where thefe two Circles of yo.
and 2 fhallcrofs this occule Line, there will be the Centers for the Houre _
circles of rm and 7;and sand a. Again, divide the diftance between the.
Centers of 10 and 2 intothree equal parts, fo the feet of the Compafles:

> Having three points E, P, W, find their Center, which will fall in the.

_willreft in cwo points; the one is the Center of the Hour. circle of 8, and”

the other the Center of the Hour-circle of 4.; and theextent of che Comes

_ paffes co one of thefe chird parts fhall be the true Semidiameter of thefe-

Circles, if there be.no crror committed in the finding of. the other Cen=
ters, .

The Hour-circles being thus drawn, take* 51 gr. 30 m.from C toward’
W, and prick them down in the South part of the Meridian from C unte-
A, and bring the third point E A’W into a Circle, chis Circle fo drawn.
fhall reprefent the Equator. no

The Tropick of & is 23 gr. 30 m.above the Equator, and 66:gr. 30 mi
diftant from the Pole, and fo in chis Laticude ic will crofsche South part’
of the Meridianat 28 gr. from the Zenith, and the. North part of che Mé-
ridian at 15 gy. below the Horizon. Take therefore 28 gr. from C toward
W, and prick chem down inthe Meridian from C unto L, fo have you the ™
South Interfe@ion. Then lay the Ruler tothe point E, and 15 gr, in the
Quadrant NE, numbred from N toward E, and note where it croffeth che |
Meridian, fo thall you have the Norch Interfeétion. . The half way. bee.
tween thefe ewo Incerfeions will fallin the Meridian at the point aaa ay.

_and the Circle drawn on the Center a, and Semidiameter a L, thall repre:

fent the Tropick of &, and herecrofs the Horizon before 4 in the morne
ing, and after 8.in the evening, abour 40 gr. Nerthward from E and W;
according to the Rifing and Setting of the Sun at his entrance into @, a
The Tropick of vpis 23 gr. 30m. below the Equacor, and 113 gr) |
30 wm. diftant from the North Pole, fothatin chis Laticudeitcrofeth che

( He Noxth: part

of the Meridian act 62 gr. below the Horizon. Take. therefore 75 rs
from toward W, and prick them down in the Meridian te |

ae

io 8 Gade et ee, oe eS DPR, «test 2 ee &
Be AROS ee Se To Pe LL Se Fe -

(=

fo have you the South InterfeQion ; then lay the Ruler to the point E,

and 62 gr. in the Quadrant NE numbred from N toward E, and note
where it croffeth the Meridian, fo fhall youhavethe North Interfection. .
Thehalf way between thefe wo Interfetions fhall be the Center whereon
| you may defcribe the Tropick of vp, and this Tropick will crofs the Hori- ~
- zon afier 8:inthe Morning, and before 4 in the Evening, about 40 gr.
| Souchward from E and W, according to therifing and fetcing of the Sun.

at his entrance into yy.

Zo find the Inclination of any Planes

~ Por the diftinguifhing of thefe Planes, we may find whether Hey be.

Horizontal, or Vertical, or inclining to the Horizon, and how much they

"incline, . either by the ufual Inclinatory Quadrant, or by fitting a Thred-
| and Plummet unto the Sector, .

For let the Seétor be opened to a Right Angle, the Lines of Sines to an.
Angle of 90 gr. inward edges of the Sector to 99 gr. and let a Thred and
Plummet be hanged upon a Line parallel to the edges of one of the Legs,
fo chat Leg thall be vertical and the other Leg parallel to the Horizon, .

If the Plane feem to be vertical (like the Wall of an upright Building):
you may try it by holding the Sector, fo thatthe Thred may: fall upom his

-Plummet-line: For chen if the verticaledge of the Sector {hall lie clofe to
_the Plane, the Plane is ere&, and therefore faid to be vertical; and if you.

draw a Line by that edge of the Sector, it fhall bea Vertical Line.

If the Plane feem to be level with the Horizon, you may try it by fee-
ting the Horizontal Leg of the Se€tor to the Plane, aid holding the other:
"way foever you turn the Sector, it is am Horizontal Plane.

If the one end of the Plane be higher than the other, and yet not verti-.

gal, it isan inclining Plane, and you may find the Inclination. in this.

manner. | A AN
Fafthold the Vertical Leg of the SeQor upright, and turn the Hori-
i Legabout, until it lie clofe with the Plane, and the Thred fallon.

his Plummet-line; fo the Line drawn-by che edge of that Horizontal Leg

thall be an Horizomal Line.

* Suppofe the Plane to be BG ED, and that B D were thus found to be

the Horizontal Line upon the Plane, chen may you orofs the Horizontal
Line at Right Angles with a Perpendicular CF: that done, if you fet
one of the Legs of the Seétor.upon the Perpendicular Line'CF, and

make.

Leg upright: For then if the Thred fhall fall on his Plummet-line,which ;

To fad the Inclination of a Plane. Uy:

a

ME SOs ch PUI Ry to at ey 8
Be ~ : : - : F ‘ aie ines é sage Oe ¢ Ses ee ape! EN | .
: “6 Fo find the Declination of a Plane;
+ . F ° 1 ; A
F _spemensaae, ~ 2 soe
: —
a's ————
q en
i 4) 53 : . E
as EB
4 (3s si ==
. i = 2S
H — ==
i = (==
; ihe = =
y <= pe
¥
: =
ft
#
\
oes oi .
Ay By : ; : ste
is ig mné ATT ae
a ae SS —— haat Ae ‘ = = if

make the other Leg with a Thred and Plummet to become Vertical, you |

_- ment of chis Angle isthe Inclination of the Plane co the Horizon.

«@.
ay |

f mts

To find the Declination of a Plane.

The Declination of a Plane is always reckoned in the Horizon between ~

the Line of Eaft and Weft, and che Horizontal Line upon the Plane, As

inthe Fundamental Diagram, the prime Vertical Line (which is the Line

of Eaft and Weft) is ECW if the Horizontal I.ine of the Plane pro-
' pofed fhallbe BC D, the Angle of Declination is EC B. mae
But becaufe.a Plane may decline divers ways, that we may the better dis
ftinguith them, we confider three Lines belonging to every Plane: tae

‘ | 5

— ———= ‘da, 4 d ff ‘|
hy

Ai fhall have the Angle between the Vertical Line and the Perpendicular on |
a: the Plane, as before in the Ule of the Sector, pag. 62. and the Comples |

|

i

}

|

lee

firftis the Horizontal Line ; the fecond, the Perpendicular Line, crofing
the Horizontal at Right Angles; the chird, the Axis of the Plane, croffing.
both the Horizontal Line, and his Perpendicular, and the Plane ic felf ac
‘Right Angles. .

) The Perpendicular Line doth help to find the Inclination of the Plane,

denomination unto the Plane.
_ Forexample: Ina Vertical Planein the Fundamental Diagram, repre-
fented by F ZW, the Horizontal Line is EC W, the fame with the Line
of Eaftand Weft, and therefore no Declination. The Perpendicular cof.

fing ic is C Z, the fame with the Veriical Line, drawi®from the Center to.

1

Axis of the Plane is SC N, the fame with the Meridian Line, drawn from.
the South tothe North, and accordingly givesche denomination to the-
Plane. For the Plane having two Faces, and the Axistwo Poles, S and:
Nj the Pole S fallmg:direétly into the South, doth caufe chat Pace to
which it is next, to be called che South Face; and the other Pole at N,
pointing into the North, doth give the denomination to the other Face,.
id make ic to be called the North Face of this Plane,

Diagram, reprefented by BF D, the Horizontal Line is BC D, whicto
Iecech the prime Vertical Line ECW, and therefore it is called
Declining Plane, according to the Angle of Declination EC B ov.
NCD. The Perpendicular to this Horizontal Line is CF, where the
int F fallechin the Plane QZ H perpendicular to the Plane propofed;
1 @ . : . :
vetween the Zenith and the North part of the Horizon $ and therefore ie
ow a Plane inclining to the Norchward, according to the Ark F

nthe AngleF CQ. The Axisof the Plane is here reprefented by the .

tne C K, where the Pole Kis 99 gr. diftant from the Plane, and (0 isas
nuch above the Horizon at H, and the other Pole as much below the Hoe
‘iwonai Q, asthe Plane at F is diftant from the Zenith: And this PoleK
ere falling between the Meridian and the prime Verrical Circle into the:
outhweft pare of the World, this upper Face of the Plane is cherefore
alled the South-weft Face, and che lower the North-eaft Face of the
dane. :

The Declination from the prime Vertical may be found by che Needle
athe. ufual Inclinatory Quadrant, or rather by comparing the Horizon-
Line drawn upon the Plane; with the Azimuth of che Sun, and the
feridian Line, in fuch fort as before we found the Variation of the. see?
hott ese Nie metical |

i To find the Declination of 4 Plaiied =
as before; che Horizorftal, tofind the Declination; the Axis) to give. -

theZenith, right unto the Horizon, and therefore no Inclination. The: -

Ta like manner, in the Declining Inclining Planein the Fundamental’

Thread falleth on B D or HO, and then a Triangle refolved, the Declina+

$F finde Deckntsion ofa lave? =
netical. Needle. For take any Board that hath one fide ftraight, and

draw as in the laft Diagram the Line H O parallel to that fide, and the

‘Line Z M perpendicular untoit, and onthe Cenrer Z make a Semicircle
~ HMO: this done, hold che Board to thePlane, fo as H © may be paral-

lel co B Dthe Horizontal Line on the Plane, and che Board parallel to the
Horizon: thea the Sun thining upon it, hold outa Thred and Plummet,
foasthe Thred being Vertical, the thadow of the Sun may fall on the
Center Z ; and dyaw the Line of Shadow A Z, reprefenting the common
Section which the Azimuth of the Sun makes with the Plane of the Ho-
rizon, and let another take the Altitude of the Sun at the fame inftants
fo by refolving a Triangle, as fhewed before, you may find what Azi-

muth che Sun was in when he gave fhadow upon A Z,

Suppofe the Azimuth tobe 72.¢7. §2 m. from the North to the Weft-
ward,and therefore 17 gr.8 m. from the Weft, we may allow thefe 17 gy
$ m.from A unto V,and draw the LineZ V,and fo we have the true Welt
point of the prime Vertical Line: then allowing 90 gr. from V unto $,
we have the South point of the Meridian Line ZS, and the Angle HZ V
fhall give the Declination of the Plane fromthe Vertical, andthe Angle
OZS the Declination of the Plane from the Meridian. |

Or wemay take out only the Angle A.Z H, which the Line of Shadow
makes with the Horizontal Line of the Plane, and compare it with che
Angle AZ V, which the Line of Shadow makes with the prime Vertical,
And fo here, if A ZV the Suns Azimuth fhall be 17 gr. 8 m. paft the
Weft, and yee the Line of Shadow A Z 7 gr. 12 m, {hort of che Plane, the
Declination of the Plane thall be 24 gr. 20 m, asmay appear by the fite
of the Plane and the Circles. oad

If the Altitude of the Sun be taken at fuch time as the Shadow of the |

tion of the Plane will be fuch asthe Azimuth of the’Sun from the Prime

Vertical. : 5 |

i
f

Tf at fuch a time as the Shadow fallech on MZ, the Declination wall be
fuch as the Azithuthof the Sun from the Meridian, |

Tf it be a fair Summers day, you may firft find what Alcieude the Sun.
will have when he cometh tothe due Eaft or Weft, and then expe& watil

~ hecome co that Altitude, fothe Declination of the Plane fhall be fuchat

the Angle corttained between the Line HO and the Line of the Shadow. |
_ Having diftinguithed the Planes, the next care will be for the placing of.
the Style, and the drawing of the Hour-lines. 7 STH |
The Style will be as the Axis of the World, fometimes parallel re

| ssi Plan

ee. |

a. |

ip

2 20

The Hour-lines will be eicher parallel one to the other, or meetin a

Center with equal Angles, or meet with unequal Angles. If the Style
_ be perpendicular to the Plane, the Angles at the Center will be equal;

and this falls out only in the Seuth and North Face of the Equinoétial ~
| Plane. If the Style be parallel to the Plane, che Hour-lines will be alfo

_ parallelone to another; and this falls out in all Polar Planes, asin the Eaft
| and Weft Meridian Planes parallel to the Circle of the Hour of 12, in che
- upper and lower direct Polars, parallel to the Circles of che Hour of 6,

| andin the upper and lower declining Polars, which are parallel to any of

| the other Hour-circles.

| But in the Horizontal and all other Planes, the Style will cut the Plane
| with an acute Angle, and the Hour-lines will meet at the roor_of the
| Style, and there make unequal Angles. |

CHAP, I. ;
To draw the Hour-linesin an Equinottial Plane.

|

| ,
A N Equinodtial plates that which is parallel to the Equinoétial
|

. Circle here reprefented by E A.V, wherein che {paces between the

cara

| Hour-circles being equal, there ts no need of further Precept, but only to
| draw a Circle, and to divide itinto 24 equal parts from the 24 Hours,and

fabdivide each Hour into Halvesand Quarters, and then to fer up the

H

Style petpendicular.to che Planein the Center of the Circle. “The help
| which thefe Lines of Proportion.do here afford us, is only in the divifion |

of the Circle, which may be done readily by chat which I thewed before
in the Firft Book of the Seftor.
| For Example: Suppofe the Semidiametér of the Equinodtial Circle to
Defix Inches, and chat ic were required to know the diftance of the Hour-
| points each from other; here each Hour being rg gr. diftant from other, [
extend the Compafies from the Sine of ‘50 gr. unto the Sine of 7g. 30 m.
the half of 15 gr. and I find the fame extent-to reach in the Line of.
/Numbers,from 6. oounto 1.56,
| Orin crofs work I extend chem from che Sine of 30 gr. unto 6.00 in
the Line of Numbers, the fame extent will reach from the Sine of 7 gr.
| 30m, unto 1,56 in the Line of ens which fhews that in a ae
B ot
|

The Defeription of the Hour-lines in an Equinottial Plane, “9
! i fometimes perrendicular, fometimescut the Plane with Oblique
| &ingls, ;

of fix. Inches femnidiameter, che diftance of the Hour-points each from.
other will be about 1 Inch and 56 Cente/ms.or partsof 100, The like rea-.
fon holds for the infcribing of all other Chords in the Prop. followings ©

CHAP. II...
To draw the Hour-lines in a Direét Polar Plane.

_ Dire& Polar Planeisthat which is paralleltothe Hour of 6,: and

here reprefented by EP W; wherein the Style will be parallel to

the Plane, and the Hour-lines parallel one tothe other 3 and therefore

may be beft drawn by that which I have fhewed in the Ufeof the Seéfor.

_ They may be alfo drawn by the help of thefe Lines of Proportion, in this
manner,

Firftdraw aRight Line W E for the Horizon and the Equator, and

crofs

| 00 .& Polar Plante —

“crols it at the Point-C, about the middle of the Line, 3
with C B another Right Line, which may ferve for _

| the Meridian and the Hour cf #2, and muft allo be the
| Subftylar Line wherein the Style thall ftand. Then, to
| proportion the Style unto the Plane, confider the length of
the Horizontal Line, and what Hour-lines you would
have co fall on your Plane. |
For the diftance of any one Hour-line from the Meridi-
an being known, we may find both the length of the
Style, and the diftance of the reft: becaufe,
As the Tangent of the Hour given,
Is tothe Diftance of the Meridian :
Sothe Tangent of 45 gr. |
To the Height of the Style.
Suppofe the length of the Horizontal Line to be 12 In-

ches, and that ic were required to put on all the Hour-lines —

from 7 in the Morning unro 5 inthe Evening. Here we
have 5 Hours and 6 Inches in either fide of the Meridian :
Wherefore I allow 15 gr. for an Hour, and extending the
Compaffes from the Tangent of 7§ degrees, I find the

fame extent to reach in the Line of Numbers from 600 to

| the diftance of the Hour-points of 9 and 3 from the Meri-
dian, to be rInch 6r parts. |

To find the length of the Tangent between the Sibftylar
ip and the Hour-points.
| eAs the Tangent of 45 gr.
Tothe Tangent of the Hour :
Sothe Height of the Style
To the length of the Tangent-line between the Sub-
Stylar and the Hour points.
Thus having found the length of the Style in our Ex-
ample to be i. 61, if I extend the Compaffes from the
Tangent of 45 gr. unto the Tangent of 15 gr. the meafure
_ of the firft Hour from the Subftylar, I thall find the fame
-extentto reach in the Line of Numbers from 1.6 4 unto
©. 43, for the length of the Tangent between the Subftylar
and the Hour-points of rr and a. If I extend them
| ae ; B bb 2 from

~~

fabour 161, This fhews both the height of the Style, and.

bn 71.19
gol S45
| 60] 70pm

7,8

BAO E OMG Ba OM 6 AM MBA AR Me Aste she tas AMA AAMTAl UMAGA LI SIST:

6

Exza

Bi

Me

—— SSeS TC ec

eS eS A ee NSD rw OT: a ae we = = be ww
iT

2

1o

eo esew7

a RS = a

AE a 1

ba Poke Pah Be RNC EPI od SEP OL Sr omen ‘4 7

“The Defeription of the Hour-lines

rom the Tangent of 45 gr..untothe Tangent of 75 gr. the meafure of
‘he fifa Hour, I thall find thera to reach in the Line of Numbers from
1. 61 unto 6. 00, for the length of che Tangent from the Subftylar ta

che Hour-points of 7 and 5. For howfoever it be the fame diftance inthe

Line of Tangents from 45 to 7,5, as from 4¢
unto 153 yet becaufe 7§ are more, and 1§ lefs
chan 45, the Tangent Lines thac anfwer to
chem will be accordingly more or lefs than the.
| length of the Style. E |
Ik By CO 43 Again, If I extend chem from 45 gr. in the |
TO 2/30 ©8 63) Tangentsunto 30 gr. the meafure of the fecond
9 31445 % 6%) Hour, I thall find chem.reach in che Line of
8 4/60. O2 79) Numbersfrom ¥. 61 unto o, g3 for the Hour of
7 5\75 O16 © yoand2: If I extend them from the Tangent
| 6 690 olfnfin.| of 45 gr. unco the Tangent of 60 gr. for the)
fourth Hour, I fhall find them to reach in the
Line of Numbers from 4.64 unto 2.79, and fuch is the lengch-of the
Tangent Linc from the Subftylar unto the Hour of 8and 4, And the’
like Reafon holdeth for the infcribing of all other Tangent Lines in the’
Propoficions following. ei |
But for fach Tangents as fall under 45 gr. I may better ufe crofs
Work, and extend the Compafles from the Tangent of 45 gr. unto 1.61

Gr. M. In. Pa.
Iz }© ©o. Oo

inthe Line of Numbers, fo thall I find the fame extent to reach from

3ogr.inche Tangents, to 93 parts in the Line of Numbers, for the d -

_ ftance of thefecond Hour; and from tg5 gr. inthe Tangents, to 43 pasts

for tie diftance of che firft Hour from the Meridian.
; Or

: ina Meridian Planéee 13.
Orif this extent from 45 gr. backward.to 1. 61 be too large for th®
~ Compatfies, I may extend them forward from the Tangent of § gr. 43
_ to 1. 6t partsin the Linesof Numbers, and the fame extent thall reach
from 15 gr. in the Tangents, to 43 partsin the Lines of Numbers, for
| thediftance of the firft Hour; and from 30¢r. to 93 parts, for che di-.
| tance of the fecond Hour, as before.
Having found the length of the Tangent Lines in Inches and parts of
~ Inches, and pricked them in the Equator on both fides of che Meridian,
- from the Center C3 if we draw Right Lines through each of thole
Points, crofling the Equator at Right Angles, they fhall be the Hour-
Lines required ; and if we fer a Style over the Meridian, fo as the edge of
it be parallel co the Plane, and the height of it be as much above the Me-
ridian, asthe diftance becween the Meridian and che Hour-points of 3 and
9, ic hall reprefencthe Axis of the World, and be truly placed for the
cafting of che Shadow upon the Hour: lines in a Polar Plane.

—

3A | The Defcription of the Hour-lines

CHAP. HI.

To draw the Hour-lines in a Meridian Plane.

ra

A Meridian Plane is that which is parallel to che Meridian Circle in

; the Fundamental Diagram, reprefented by $.-ZN 3 it hath two
‘Faces, one to the Eaft, and the other to the Weft; ineach of them the
“Style will be parallel co the Plane, and the Hour-lines parallel one to the

N

cee scogeredracUREURGeenaurMNNeCeUyCD Kekksae acoso
a a 4

:

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€
[Yi

aa
s
Ky

oy

sEbe 20

ae.

sEbS a
[\

pa

= “ <
Ss
a4
cite ANS

other, asin a Polar Plane ; the difference being only in the placing of
the Equator, and in numbring of the Hours. 1
For in thefe Meridian Planes having drawn an occult Vertical Line
CZ, and an occult Horizontal Line C N, croffing onethe other at Right
Anglesia the Point C, the Equator AC will cuc the Vertical with an —

Angle

|

Paes 3 an Horizontal Plane. | 15
—Bnele ZC A, equal to the Latitude of the [EAng. Po.) Tang. |
Place: chen may wecrofs che Equator at Right [7 |Gr. M. In. Pa.
- Angles with the Line C B for the Hour of 6, RN cory
and from this fet off the Hour-points in the las Sate
- Equator, as in the former Prop. | ee
gq > i Ole Vane
Fot, {uppofing the length of the Style hee ae
CB to be ro Inches, the length of the Tan- nls aa
gent Line belonging to the firft Hour will be i8 ——_
2In. 68 p. the length of the fecond 5 14.77 )- Piao
asin the Table. Then the Tangent of 15 gr. ; oe fae
being prickt down in the Equator on both 5} + 93
fides from 6, {hall ferve for the Hours of § poh ke we
and 7, and the Tangent of 30 gr. for the 45} © 68
“Hours of 4 and 8; and fo in the reft. 37. 301 7 07
This done, if we draw Right Lines through 15) 8 77
each of thefe Points,. croffing the Equator at OT SEROS
‘Right Angles, they fhall be the Hour-lines 48 45/12 40
“required : And if we fet a Style over the O13 03]
“Hour of 6, fo as the edge of it may be pa- I5|14 97].
“allel co che Plane, and the height of ic may OOF OP sae
be equal to the diftance between the Hours : 45\20' .28
of 6 and g.in the Equator, ic fhall repre-. | (97 30/24 14
‘fent the Axis of the World, and be truly 29 46
placed for the cafting of the Shadow upon 037 32
‘the Hour-lines in a Meridian. Plane. , 45 50 27
| 30,75 96
1S bed 57
Ollinfinit,

CHAP. IV.
To draw the Hoar-lines in an Horizontal Plane

-N Horizontal Plane is that which is parallel to the Horizon, repre -
“fented in the Fundamental Diagram by the outward Circle

| ESWN, in which the Diameter SN drawn.from the South to the -
| North, may go both for the Meridian Line and the Meridian Circle, Z °
for che Zenkh, Pfor the Pole of the World, and the. Circles drawn:

through. ;

PN ee ee
. 4
Zs

Nn

16 The Defeription of the Hour-linss ts
Tait: 51 30 through P for che Hour-circles of 1, 2, 3,4, Gc,

| TlAng.Po.JArePla,'. a) they are numbred from the Meridian. 3
° (Gr. M.|Gr, M.| _ Thefe are equal at the Pole, and at the Equa-

72}0 of oof for, but unequally diftant at the Horizon the

3. 45| 2-56 diftance between the Meridian and the firft Hour ~
| 7 30 z being not full 12 gr. the diftance between the
ih as 3 st fifch and fixch Hour about 18 gr. which inequa-

| lity being obferved, if you fuppofe Right Lines
3 drawn from the Center C to the Interfections of |
19 49)14 54) chefe Hour-circles with che Horizon, the Line fo
| [FE 3°87 57) drawn fhall be che Hour-lines here inquired.
Bag And then, if you can, imaginea Line drawn |
; ede 0124 20 from the Center C, toward P the Pole of the
33 45127, 36) World, and raifed above the Meridian Line |
© CN, foas the Angle P C N may be equal to the ©
41 15134 28) Yoticudeof thePlace, this Right Line C P hall |
3\45_ 0/383) be the Axis of the Style. And foyouhave both |
| 148 45/40 45] Styleand Hour-lines ready drawn to your hand, »
$2 39/45 34, But more particularly to our purpofe. |
56 15149 30 Thefe Hour- circles confidered, with the Meri- -
4160 —©}53 3§| dian and the Horizon, do make divers Triangles, |
63 451§7. 47| PN3i,PN2,PN 3, in which we have known,
67 30/62 .6| firft, the Right Angle acN, the North Interfes
7i 15/66 33) tion of che Meridian and the Horizon; fecond-
5|75.O|7" 6} ly, the Side P.N, che Ark of the Meridian bes
17178 45/75 45] cween the Poleand che Horizon) which is always |
25} equal tothe Latitude of the Place; thirdly, the.
186 15|25 13] Angles atthe Pole, made by the Meridian and |
o} the Hour-circles, the Angle N P x being 15 gr, |
N P 2 30gr. each Hour 15 gr. more than orher, |
each half Hour 7 gr.30 m.cach quarter 3 gr.45 7. as in the fecond Column’
of this Table. ‘And thefe three being known, we may find the Arcs of |
the Horizon between the Meridian and the Hour-circles N 1, N 2, N 3,
: OC. For, |

ett the Sine of 90 er.
~ dstothe Sine of the Latitude :
So the Tangent of the Hour,
to the Tangent of the Hour-line from the Meridian.

Extend

ka The Defeription of the Howr-tines eer
| Extend che Compaffes from the Sine of 90 gr. to the Sine of the Lati-
| tude; fothe fame Extent fhall reach from the Tangent of the Hour, to
: | ;

the Tangent of the Hour-l'ne from the Meridian. Thus the Laricude-be-
ing 51 gr. 30 m. I extend the Compaffes from the Sine of go gr. tothe
Sine of 51 gr. 30 m. and find the fame extent to reach from the Tangent
of 3 gr. 45 m. unto the Tangent of 2 gr. 56 m. for the diftance of the
. fire quarcer from the Meridian ; and from the Tangent of 7 gr. 30 mm.
unto the Tangent of § gr. 52. for the half Hour; and from the Tan-
gent of al gr, 15 m.to the Tangentof 8 gr. 51 m. for the third quarters
and from the Tangent of 15 gr.Om. unto IE gr. 50 m. for the firft
Hour: And foof chereft, asin the third Column of this Table, under
the Title of the Arks of the Plane. 3
Cce “Only

~

iene in an Horizontal Plane,

weweerses * i SS x1 =e aia: fae AES | vt TT tow em

nN Roe
iN

Only when FE come to fet one Foot of the Compaffes to 48 pi 45-Ms
for the finding of a quarter paft 3, the other Poor will fall out of the
Line, and then I may either take out fo muchas is out of the Line beyond

45 gr. and turn it back into the Line, and it will reach from 45 Bre tO

4i gr. 45m. or Y may ule crofs work, extending the Compaffes from

the Sine of 90 gr. to the Tangent of 48 gr. 45-m. fothe {ame extene will —

reach from the Sine of §1 gr. 30m. to the Tangent of 4t gr.45 m. And

fuch is the diftance of the Line of 3 Hours + from the Meridian.

This done, I eometo the Plane, and there according asthe Lines do

fall in the Fundamental Diagram,

1, Idrawthe Right LineSN; {erving for the Meridian, the Hour |

vof 12, and che Subftylar.

' 2 |

oe ole) wee tee oe ee Ste Se
it aes
na b Te
sf fo =
: aa

Wh AVertical Planes je “Ep
2, In this Meridian I make choice of a Center at C, and there de-
- feribe an occultCircle reprefenting the Horizon. a
3. [find a Chord of 11 gr. 50 #5, and infcribe it into this Circle on
_ either fide of the Meridian, for the Hoursof rrand 13 in like manner,
_aChord of 24 gr. 20 ws. for the Hours of 10 and 2; and a Chord of
38 gr. 3 ws. for the Hours of 9 and 3: And fo for the reft of the Hours,
_ their Halves, and Quarters,
4+ Idraw Right Linesthrough the Center, and the Terms of thefe -
_ Chords, and thefe Linesfo drawn arethe Hourelines required.
The Line belonging to the Hour of 6 willbe perpendicular to the Me-
fidian, and the Hour-lines before 6 in the Morning, or after 6 in the
Evening, may be fupplied by continuing their oppofite Hour-lines beyond
the Centers as the Hour-line of 7 in the Morning continued, will be the
‘Hour-line of Zin the Evening: And {othe reft.
__ Laftly, T fet up the Style overthe Meridian, fo as it may cut the Plane
inthe Cencer, and there make an Angle with the Meridian equal to the
‘Lasitude of the Place; fo it thall reprefent the Axis of the World, and -
be truly placed for cafting of the Shadow upon the Hour-lines in an Hori-
zontal Plane. | |

a3
| ee
‘a

i CHAP. V.

To draw the Howr-lines in a Vertical Plane.

A. Vertical Plane is that which is parallel to the Prime Vertical
A Circlein the Fundamental Diagram, reprefented by EZ W. Ie
hath two Faces, the one to the North, the other co the South; in each of
them the Subftylar will be the fame with the MeridianLine,and the Angle
of the Style above the Plane will be equal to Z P,the Complement of the
Latitude ; and the Hour-lineshere inquired may be fupplied by imagi-
ning Right Lines drawn fromthe Center C to.the Interfections of the
Hour-circles EZ W. | , ;

_ The Triangles here confidered are made by the Vertical, the Meri-
dian , and the Hour-circles, in which we know the Side ZP, the
| Ocee Angles

Se

NG pers

20 " The Deferipsion of the Hour: lines

Angles at the Pole, and the right Angle at the Zenith, and cherefore
may find the Arks of the Vertical, becween the Meridian and the -
Hour-cireles, afcer this manners a |
eAs the Sine of 90 er. a
is to the Co-fine of the Latitude: |

So the Tangent of the Hour,

to the Langent of the Hour-line frems the ~Miridian:
Excend the Compaffes from the Sine of 90 gr. to the Sine of the
Complement of the Latitude, fo the fame extent fhall reach from the
Tangentot the Hour, to the Tangent of che Hour-line from the ea

dian, Thus

rend the Compaffes from ‘the Sine cf 9° gr. to-
_ the Sine of 38 gr. 30 m. and find the fame ex-

gr. 42m. for the fifth Honr « and fo in thereft, | ales 9 28

= ec SE ESS

4 do fall in the Fundamental Diagram... Bie Aha a
78 45/72 17

62. 36179. 43

a1 be By 86 1§ 84. Cc

“matt bein the lower part of the Meridian Line, | 4 60 147° 9
and the Style point upward in all {uch Places as 63 45151 36
~areto the Northward of the Equinoctial Line, . 67 30156 20
"as it- may appear by confidering how the Lines. | 7! 35/61 33

da in aVertical inclining Plane. ek eee

“Thus inthe Latitude.of 51 gr. 30m Tex- (Latirs 9 51 30) ,
rc Ang. Po.|Arc. Pla.)

Gt M.iGr. M.

| _ tent to reach trom the Tangent of 45 gr- (0 |12} 0 | 0 o
the Tangent of 9. gr. 28 730 for the diftance of hte 3 urs 5| 2 |
the. firtt Hour from the Meridian3 and from “7 Me ‘s
the Tangent of 75 gr. unto the Tangent of 66 tieagds i | «.

as in this Table. |

| Thefe Arks being known, I may come to the ny 45/IT 56
' Plane, and’ then by help of a Thread and i ‘ ZO;14 29
Plummet draw a Vertical Line, ferving both for 1 , i IS|IF+ 4
O'9. 45

the Meridian and the Hour of 12, and the Sub- =—/————

-ftylar 5 then may I draw an occult Vertical 33 45/22 35
~ Circle, and therein infcribe the Chords of thofe 373/25 32

former Arks, and draw the Hour-lines, and fet 40 15128 38
~ up the Style, as before in the Horizontal Plane. 345 0/3854)

If ic be the South Face of che Plane,. the Cen- 43 45135 22
cer will be upward, and the Style will point $2 30/39 3
‘downward : If the North Face, the Center | 156 15/42 58

|} 6190 e990 c

CHAP. VIL

Ty draw the Hour-lines in aV ertical Inclining Plane.

LI thofe Plancs that have their Horizontal Line lying Eaft. and
| Weft, are in that refpect faid to be Vertical ; if they be alfo up-
right and pafs through the Zenith, they are dire&t Verticalss. if chey -

“incline tothe Pole, they are dire&t Polars; if to the Equinogtial, they .

aye.

LARA TC A OER Rhy MERE ORE TY AYES? SOREN eo) ell ee a

22 The Defeviption of the Hour-lines ;
are properly called Equinoétial Planes, and are defcribed before: if to

none of thefe three Points, they are thea called by the general name of
~Tnclining Verticals.

Thefe may incline either to che North parts of the Horizon, orto the
South ; and each of them hath two Faces, oneto the Zenith, the other
tothe Nadir, in which we are firftto confider theheight of the Pole

\

- above the Plane, by comparing the Inclination of the Plane to the Hori- —

zon with the Latitude of the Place. } |
As in our Latitude of 51 gr. 30 m. if the declination of the Plane

EI W inthe Fundamental Diagram fhall be 23 gr. Northward, thacis,

if IN, the Arkof the Meridian between the Plane and the North part

of the Horizon, fhall be 13 gr. we may take thefe 13 gr. out of PN 51 gr, |

30 ms. the Elevation of the Pole above the Horizon, and there will remain
P 138 gr. 30 m. for the Elevation of the North Pole above the upper Face
of the Plane, and therefore 38 gr. 30 m. for the height of the South Pole
above the lower Face of the Plane. f

Or if the Inclination of the Plane fhall be found to be 62 gr. tothe —

Southward, wemay number them in the Meridian from S the South pare

ef the Horizonunto L, and there draw the Ark EL Wreprefenting this

Planes fo the Ark of the Meridian P L fhall give theheight of the North —
Pole above the upper Face of this Plane to be 66 gr. 30 m. and therefore —
the height of the South Pole above the lower Face of the Plane is alfo

66 gr. 30 m. : G
In like manner, if the Inclination of the Plane EY W (hall be Ig or
Southward, that is, if S Y che Ark of the Meridian between the South

part ef the Horizon and the Plane fhall be 15 gr. cheheight of the North |

Pole above the upper Face of the Plane, and the height of the South Pole

above the lower Face of the Plane, will be alfo found to be 66 L230 7.

But if the Plane fhall fall between the Zenith and the North Pole,
then will the North Pole be elevated above the lower Face, and the South
Pole above the upward Face of the Plane, as may appear by che Projecti-
on of the Sphere in che Fandamental Diagram,

Then in the Triangles made by the Plane, the Meridiag , and the >

Hour-circles, we have the fide which is the ‘height of the Pole above
the Plane, together with the Angles ar the Pole, and the Right An-
gle at the Interfection of the Meridian with the Plane, by which we
may find the Arks of the Plane between the Meridian and the Hour-
circles, after this manner. f

es

=? . cs ~

in a Vertical Inclining Planes =a

As the Sine of 90 er. | ahs
Is to the Sine of the Pole above the Plane:
So the Tangent of the Homr, |
To the Tangent of the Hour-line from the (Meridian.

Thus in the former Example, where P I the heighe of the Pole.
above the Plane was found to be 38 gr. 30 m. if you thall excend
the Compafies from the Sine of 90 gr. to the Sine of 38 gr. 30 m.
the fame extenc will reach from the Tangent of 15 gr. unto the
Tangent of 9 gr. 28s. for the diftance of the firft Hour from the
Meridian, and from 30 gr, unto 19 gr. 46 m. for the fecond Hour, and.

fo forward, asin che dire Vertical.

__ And for the two laft Examples, you may extend the Compaffes from
‘the Sine of 90 gr. unto the Sine of 66 gr. 30 m. fo tke fame extent hall
_ reach in the Line of Tangents from 1 gr. unto 13 gr. 48 ms, for the firft

Hour, from 75 gr. unto 73 gr. 43 m. for the fifth Hour, from 30 grs-
unto 27 gr. §4m. for the fecond Hour, from 66 gr. unto §7 gr. 48 ms

_ “Thefe Arks being known, you may firft draw the Horizontal Line;

vand crofs ic in the middle with a Perpendicular, chat may ferve borh
for the Meridian and the Hour of 12, and the Subftylar; then know-

ing which Pole is elevated above the Plane, you may accordingly.

-makechoice of a fic Point in the Meridian for the Center of your Hour-

lines, and thence defcribe an occule Ark of a Circle, inferibe the Chords

_of thofe former Arks, and draw the Hour-lines, and fer up the Style, as.

I fhewed before in the Horizontal Plane.

CHAP,

for the fourth Hour, and from 45 gr. unto 42 ¢r.31 m. for thethird —
Hour from the Meridian. !

The Defeription of the Hour-lines

CHAP. Vil.
To draw the Hour-lines ina Vertical Declining Plane.

nmay draw a Vertical Line, aresn _

-LI upright Planes whereon a mann ?
this refpect faid to be Vertical; if they thall alfo ftand dire@ly Eaft-

and Welt, they are direétly Verticals; if dire€tly North and South,

an Planes, and are defcribed before: if —

they are properly called Meridi
they behold none of chefe four
fland between the prime Vertica

_ by the general name of Declining Verticals,
Thefe have two Faces, one to the South, t
which may be diftinguithed in thefe Northern
this mannere If the Sua coming to the Meri

principal Parts of the World, but thall
land the Meridian, they are then called

hz other to the Northward,
arts of the World after
ian fhall fhine upon the

Plane, it isthe South Faces if not, it is the North Face of that Plane.

Again, If che Sun hall thine upon the Plane at High-noon, and yet
longer in the Forenoonthan in the Afternoon, it is the Souch-eaft Face;

if longer in the Afternoon than in-the Forenoon, it is che South-welt
Face of the Plane. «But how much the Declination cometh to, is belt: |

When the Declination isfound, there be fourthings more to be conifts
wecan cometo the drawing of the Hour-lines.

-1,: The Meridian of the Plane, and his Inclination to ¢

2. The Height of the Pole above the Plaae.
Diftance of che Subftylar from the Meridian Line.

4. The Diftance of each Hour-line from the Subftylar.

ay

|
|
j
'

And thefe four may all be reprefented in the Fundamental Diagram, |

as in this Example, |
ur Latitude of 54 gr. 30 m. Northward, the Declina |
ght Plane fhould be found tobe 24 gr. 20 ™, |
ePRZ weknowthe Angle at RrobeaR
s the Complement of the Declination ; and

And thefe three |

~ In the Triangl
and the Angle at Z, for ici
the Bafe PZ, for it is tae Complement of the Latitude.
being known, we may find the other Angle RPZ, which isthe
of Inclination between both Meridians.

.

- ts 4 Vertical Declining Blawes ot

eg ae ee

7
be $
°e
SS
°
s*
sf 9
sot

oj es F s

i)
%|

As the Sine of the Latitude
Is to the Sine of 90 gr.
So the Tangent of the Declination
to the Tangent of Inclination of Meridians.

_ Thus in our former Example I-extend the Compafies from the Sine of
the Latitude 51 gr. 30 m. unto the Sine of go gr. the {ame extenc will
each in the J.ine of Tangents from 24 gr. 20 ms. the Declination given, to
about 30 gr. and fuch is Z PR, the Angle of Inclination between the
Meridian of the Place and the Meridian of the Plane; and thererefore
mS : Ddd che

Ba

wee:
ade es

me

ap raneiagielenniiin:

bo i Ga Nis ne ee Sere pa: Satie Rae ie ah andre Rg

the Meridian of the Plane will here fall upon the Circle of the fecond’

Hour fromthe Meridian of the Place (as ic may alfo appear by opening. —
~ the Compaffes to rhe neareft extent between the Pole and Plane) agit chere

I place the Lecter R to make chis ReGtangle P RZ..
2. To find the Height of the Pole above the Plane.

The Height of the Pole is to be meafared in the Meridian of the Ptaned
it isherereprefented by the Ark PR, and may be found by, that which,
we ‘Shp known.-in the former Triangle PRZ,

¥ As the Sine of 90 gf.
by the Co-fine of the Latitudes 7
So the Co-fine of the Declination, ol
to the Sine of the Height of the “Ax above the Plane: vr Se

Extend the Compaffes from the Sine a 9° gr. unto the Sine of 38: ri '
30.%. the Complement of the Latitude, and the fame extent will reach:
from theSine of 65 gr. 40m, the Complement of the Declination, unto.
the Sine of 34:¢r. 33 mm. .

_Or if you pleafe comake wfe of the Angle of the Inclination of. che

~ two Meridians, the proportion will’hold;.

‘As the Sise of 90 gr.

to the Co-fine of the Inclination “se CMeridlans :
So the Tangent of the Latitude,

to the Tangent of the Htight of the Pole above the Planes a

And then you may extend the Compaffes from the Sine of 90 gr, unto.
che Sine of 60 gr: the Complement of the Inclination of the Meridians,
and the fame extent will reach from the Tangent of 38.gr, 36 m. the
Complement of the Latitude, unto the Tangent of 34.gr. 33m, and fuch
isthe Ark P R the Height of the Pole above the Plane.

3« To find the Diftance of the S ub fylar from the Meridlan:

This is here reprefented: by the Ark ZR, and may be found hy that
which we have known in the former T riangle PR ot )

His As

She VG

“to the Sine. of the Declination £.)°" 0°) © ahs
So the Co-tangent Ng tlt DANA, ie a Re ne
to the Tangent of the Subftylar from the Meridian.

: 3 +g. "¢ tT oe ¢ tae j a 3 ) HES ra oR Af .<% 3
~» As the Sine of goer. ee

Extend the Compatfes from the Sine of 90 gr. unto the Sine of 24 gr.
20m. the Declination given, and the fame extent will reach from the
Tangent of 38 gr. 30 ms. the Complement of the Latitude, unto the Tan-

_ gencof 18 gr, 8 m.and fuchisthe Ark ZR, the Diftance of che Sublty-

Tar from the Meridian.

4. To find the Diftance of each Hour-line from the Subftylar.

| a The Diftances of the Hout-lines from the Subftylar arehere reprefent=
ed by thofe Arks of the Declining Vertical belonging to the Plane, which
_ areintercepted between the proper Meridian.of the Plane and the Hours
_ circles. | | ;

| we have known, firft che Right Angle at the InterfeQtion of the proper
| Meridian wich che Planes then the Side which is the Height of the
Pole above the Plane; and thirdly, the Angles at the Pole. For know-
| ing the Angle of Inclination between the Meridian of the Plane and the
Meridian of the Place, which is always the Hour of 12, we may find
the Angle between the Meridian of the Plane and the Hour of fr, by
allowing in 15 gr. and the Angle between the Meridian of the Plane and
the Hour of 2, by allowing in 30 gr. and fo for the reft: which being
known, and fet down ina Table, we may find the Arks of the Plane
from the Subfty lar to the Hour-circles, in this manner

As the Sine of 90 ¢r.
| to the Sine of the Height of the Pole abowe the Plane:
“So the Tangent of the Honr from the proper Meridian,
to the Tangent of the Hoar-lne from the SabPylar.

__ Thusinour Latitude of 51 gr. 30 m, if the Declination of an uprigh
Plane fhall be found to be 24 gr. 20 m. from the prime Vertical, the one
Face open to the South-weft, che other to the North-eaft, I may number
| Ddd 2 thefe

To this purpofe we have divers Triangles made by the Declining:
| Plane, together wich his proper Meridian and the Hour circles, In thefe —

aes 1am

33 «= Tha Defeviphiahof the Blouv-lines

thele 24 gr. 20 #. in the Horizon of the Fundamental Diagram from E
unto B, according co the fitwation of the Plane, and there draw the Ver- ~

tical BZD, which thall reprefent the Plane propofed.

The two Poles of this Planewill fall in the Horizon at Hand Q, and

therefore the proper Meridian drawn throwzh the Poles of the Plane and

the Pole of the World muft. be the Circle HPQ, which here croffech

the Planeat Right Angles in che Pointof R, and inclineth to PZ § the.

Meridian of the Place, according to the Angle R PZ. ay

~The quantity of chis Inclination may be readily found by the Hour-

circle where the proper Meridian fallech. Ashere it falleth on the fecond

Hour-circle, and fo the Inclination is 30 gr.
The height of the Pole above the Plane, .which giveth the height of che

Style above the Subftylar, ishere reprefented by the Ark PR. For as in
che Horizontal, fo in this andall other Planes, the Line C P the Axisof _

the World isalways the Axis of the Style, and the neareft Line that can.
be drawn upon the Plane toche Axis of the World is che firceft for the

Subftylar, and thatis the Line C R: fo the Angle PCR isthe Angle |
berween-the Axis and the, Plane, commonly called the Height of che

Style, and che meafure of. this Angle isthe Ark P R. This Ark is always
le(s than the Complement of the Latitude, and may be eftimated by
caking che diftance PR with the Compaffes, and meafuring it in the Me=

ridian from Ptoward Z. Soin this Example it will appear to be about

34 Lo 2o..

The diftance of the Subftylar from the Meridian is here reprefented’

by the Ark ZR: For the Meridian Line upon the Plane is C Z, theSub-
{tylar Line is C R ; fo the Angle contained between them is ZCR, and
the meafure of this Angle is the Ark ZR, which taken wich the Com-
pafles, and meafured in the Semidiameter CW; from C toward W; will

‘be found about 18 gr.

The diftdnces of each Hour- line from the Subftylar are here reprefented
by the Arks of the Plane between the Point R and, che Interfections of
che Hour-circles:. For. the Subftylar-Line is CR, and the Hour-circle of

x croffing the Plane inthe Point O, the Hour-line of 1 upon the Plane

muft be CO ; fo the Angle between the Sudftylar and the Hour-line of

ris RCO, and the meafure of this Angle is the Ark RO. In Like . |;

manner, the Hour-line of 12 will be C Z, and the diftance from the |

Subftyfar R Z: the Hour-line of 11 willbe CX, aud the diftance from
the Subftylar RX: and forthe reft. Thele diftances RO, RZ,RX, oan

say alfo be raken with the Com paffes, and meafured as before, 4

Befides |

|

| ene ft

‘it a Vertical Declining Plane. 26

Befides thefe fourReprefentations, the Diagram will thew what Pole
is elevated above the Plane, and what time the Sun fhineth upon the.

Plane. If icbeche Norch-eaft Face of this Plane, you may think P to

be the North Pole, and the Hour-circles co be drawn on a Convex He-
mifphere; foC R the Subftylar, and CP the Axis of the Style, will
will both point upward - and having drawn the Tropick of %, you hall.
find by the meeting of the Plane wich che Tropick, and che Hour-circles,

that che Sunat che higheft may thine upon the Plane from the time of the

_rifing untilit be paft 9 inthe morning, and from 7 inthe evening unto
| the timeof his fetting. Birt if it be the South-weft Face of the Plane,
then you may either fuppofeche Subftylar and the Axis to be continued

ar

- down below the Center, like unto the Hours before and afer 6 19 an Ho-

rizontal Plane ; or elfe you may turn the Diagram, and think P to be the
South Pole, and the Hour-circles to bedrawn in an Horizontal Concave,
fo CR the Subftylar, C P the Axis of the Style, will both point down-
ward, and fo alfo the Hour-lines from 8 in the morning until after'7 in

the evening, as it doth appear by the meeting of the Plane with che Ho- ~

rizon, and the Hour-circles. » 3 |
Thus with the drawing of one Line in the Diagram, to reprefent the

Plane according ro his declination) you may have the Hour-lines fitted to,

any Declining Vertical, with the Style and Subftylar in their due places
which may fuffice to free you frora grofs errour ; but for more exadinelss .
we confider three Triangles. . : |

1s To find the Inclination of Meridians, -

~The Meridian of the Place is a Circle pafling through the Poles of the
World, the Zenich and the Nadir. The proper Meridian of the Plane

"isa Circle paffing through the Poles oftthe. World and the Polesof the
Plane. The Circle of the Plane and thefe two Meridians do make a Tris ~
angle, fuchas P RZ, wherein we know the Angle ar R..

Iconfider. the Angle of Inclination of the Meridian RP Z,..and there

fee how that PZ, che Meridian of the Place) which isthe Hour of 12,
| being 30 gr. diftant from P R the Meridian of the Plane, and chat.one-

Face of the Plane being open to the Souch-weft, and the other. to. the
North-caft, chis Meridian of the Plane falleth to be the fame with the
Hour of 2, (otherwife with the Hour of 10:) therefore allowing 15 grs
for an Hour, the Hour of 1 RPO will be 15 gr.-and R PX the Hour -:

of x1 willbe 45 gr, diftant from PR the proper Meridian of the Plane:

And |

go | | The De(cription of the Hour-lines

And fo I gather the Inclination of the

vatitude N. 5 £ 30

Declinac. 24. 20

Dis Merid. 30 0

Alc. Styl. 34 33
Oilk. Subthe 22 8s

Hours. Ang. Po,jAr. Pla.

=—-_--—_—_—_—

M. EjGr. M,|Gr.

Ts 8Igo 00/90 ,C
5 775 Coj64. 42
6 C150 00/44 30
7 S145 00/29 33}
8 4lro ocol'8 = 8
9 3it5 oof 8 38
ro =. 2} Meri. |Subftyl.

Bi Nis ool §
I2 |30 00/88.

I 45 O0j}29
2 ICl6@e 00.44
3 9175 0064
4 8lg0 oclgo

and-North-weft.

‘t. By the help of a Thred and Plummet I draw a Vertical Line, M
ferving both for the Meridian of the Place, and the Hour of 12 ae |

2, In this Metidian Line-I make choice of a Center atC, in the upper
part of the Line if itbe the South Face, as here we fuppofe ic, that the

M.

Seeheeeeeee

38)
g

ete ae

~reft of the Hour-circles towards this Me-

ridian, according to their Angles at the
Pole, as in the fecond Column of this
Table, o

Then taking. my Compafles in my hand,
T extend them from the Sine of 90 gr. unto
the Sine of 34 gr. 33 m. the height of che
Pole above the Plane, and find chem to
reach in the Line of Tangents from 15 gre
the Inclination of the Hour of rt, to 8 orm,
38m. forthe Ark of & from the Subftylar,

and from'30 gr. unto 18 gr. 8 ms. for the

Hour of 12, agreeable to the chird Prop, —
and from 45 gr. unto'29 gr. 33 m. forthe
Hour of ‘ar, and fothe reft, which Ealfo

fet down in-the chird Column of the Tas

ble.

Thefe Arks being thus found, will ferve —
for the drawiag of the Hour-lines both on »
the South-weft Face and the North-eaft Face —
of this Plane, and alfo on eicher Face of che
like Plane chat hach the fame Declination, and the Poles in che South. caft-

Style may have room to point downward-: but in the lower part of the

Line if it bethe North Face of the Plane, for there the Style muft point
upwards and upon this Center I defcribe an occult Circle,reprefenting che’

Declining Vertical belonging to the Plane.

3. I finda Chord of 18 gr. 8 m. the diftance of the Subftylar from |
the Meridian of the Place, and infcribe it into thisCircle, from the Me-

ridian unto A toward theright hand, becaufe in this Example the Meri-

dian of the Plane falls among the Hours after Noon, (for otherwife it.

muft have been infcrived toward the left hand) and there I draw the
Line C A ferving for the Subftytar. : , sae

4. According to the Table of the Arks of the Plane from the Subfty-
| | ary

* an

be _ jmaVertical Declining Plane. — i
far, T finda Chord of 8 gr. 38 m. and infcribe ic isto this Circle, from
the Subftylar toward the Mcridian for the Hourof 3. In like manner
aChord of 29 gr. 23 m. for the Hour of 11, and.a Cliord of 44 gr.
| 30m. for the Hour of 1@; and fo for the reft of the Hours, their Halfs, .
and Quarters. ; |

Too <7

ne
Ne

V Ss

|

5. Vdraw Right Lines through the Center-and the Terms of sthefe.
Chords, and thefe Lines {o'drawn are the Hour-lines required... 3
| Laftly, Tfecup the Sryle over the Subftylar, fo as it may cut the Plane _
in the Center, and there make an Angle with the Subftylar of 34.¢r.
| 33.m, according to the height of the Pole above the Plane ; foit fhall
| ee ~ reprefent :

*
ae

a 3% cue |
‘yeprefentthe Axis of t
Shadow upon the Hour-

~ rhe Defcription of the Hour lines
he World, and be truly placed for cafting
lines in this Declining Plane, © |

‘A fecond Example.

of the

<

Suppofe another upright Plane in the fame Latitude to decline from
the Vertical’65 gr. 44 m. with one Face open to the South- eaft, che other

tothe Norch-weft. Thele 65-gre40 m. would-be numbred from’E unto

~Q, and from W unto H, and the Plane reprefented by QZH: For f&
che one Pole will fall at B in che Souch-eaft, and the other-ae D in the
‘North-weft, according to the fuppofitions The proper Meridian of this.
Plane may be-fupplied by the Circle BP D, croffing the Plane in the Point
T, between the Hours of 72nd 8, and chere is the place of the Subfty-
‘Yar.. The South-eaft Face will contain all che Hours from Sun-rifing unto
-2 after Noon; and che North- weft Face, all che Hours from I after Noon

unto Sun-fetcing.
4. The Angle Z PT,

10/79
9164.
8149
7134
6|19

rf Mane eae
35 40
qO 30
14 jf
35 §6

oe
30.28
30|16
3010
30) §

5} 4. 30 2 9
Merid. Substyl.

Hours. |Ang. Po.|Ar. Pla.!
M. _E.|Gr. _M. |

Gr. M.!

Ta
16
42,

fe)
I ;

i

410 39) 2
| 3\25 30.6 58
I 2140 3012 21)
|i HS$ 30.20 28,
‘12 {70 3035 $6

1 11/8§ 3072 .§6.

Then working‘as before,
the Inclination of t

| BS
he two: Meridians,- will be

found to be about 7© degrees 30 minutes, —
2. The Ark PT, the meafure of the |
Angle: P CT, the height of the Pole above-
the Plane, and fo the height of the Style |
above the Subftylar, will be r4gr. 51 9,
3. The Ark .Z T ‘the meafure of the |
Angle ZCT, fhewing the diftance of the |

6

Subftylar from the Meridian, will be 35 gr.
-4. The Arks of che Plane between the |

Subftylar and the Hour-lines, depending
on the difference of Meridians , which is —

- here 70 gr. 30 m. or 440. 42 m. (hore of the ©

Meridian, I firft draw a -Table wich three
Columns, one for the Morning and Evening |
Hours, another for the Angles at the Pole,

and the third for the Arks of the Plane, —
and there write 7 gr. 30 a. by che Hour
of 12, and place the Meridian and*Subfty-
lar beeween the Hours of 7 and 8, accord-
ing as the Poles of che Plane do fall in the

Diagram.

Then

re tsa Wortioal Declining Plant. 3
' ~The will che Angle ac the Pole between the proper Meridian and the
Hour of rr be §5 gr. 30m. the Hour of 10 will be 4o gr. 39m. diftanc
from that Meridian; and che reft in their order : which being noted in

_ the fecond Column, the Ark of the Plane will be found tobe fuch Num-
| -bers as [have noted in the third Cclumn. m3

GE PEP Ee ae es ad ee ERE Nn as SRL RoR ey We ome wer ee 7 oeep
|
|

|
i)

| Diagram.

| A third Example, of a Plane falling near the Meridian.

After the like manner, if in our Laticudean upright Plane fhall de-

| Norch-weft, and che other to the South-eaft, we may in fome fort repre=
| fencice by the Vertical Q ZH, and then working as before,

| 4. The Angle ZPT, the Indination of the two Meridians, will: be
found to be 86 gr, 5 m. fothat P T che Meridian of this Plane will here
| fall between the Hour-circles of 6 and 7 from the Meridian. ap

_2.-The Ark P T, the meafure of the Angle PCT, the height of the
| Pole above the Plane, will be onely 3 gr, 6 ms. |

__ 3. The Ark ZT, themeafure of the AngleZ CT, thediftanceof the
| Subftylar from che Meridian, 38 gr. 23m.
4 The Table of the Angles at the Pole

willbe alfo gathered, by comparing the Me- Latitude - $830
-ridian of the Plane with the reft of the Declination 85_c
Hour-circles: For the Angle TPZ, be- |Duft. Merid. 86 5

tween TP the Meridian of the Plane, PZ Altitude Styl. 93 6
the Meridian of the Place, andthe Hour of | Dift. Subftyl. 3823
(12, being 86 gr. 5 m. allowing 15 gr. for

an Hour, the Hour of 12 £ will be 78 gr. 35 m. and the Hour of rr, ~

Jagr. 5 ws. diftanc from the Meridian of the Planes and fo che reft of
the Hours. Or becaufethe difference of Meridians 86 gr. 5 m. refolved
‘into Time, makes 5 ho. 44 m. and fo che Meridian of che Plane falls be-
tween the Hours of 6 and 7 from che Meridian, J] firft place chis Meri-
dian between thefe Hours, and then taking 75 gr. the common meafure
for § Hours, out of 86 gr. 5 ms. there remains 11 &r. § m. for the Angle |

» acthe Pole between the Meridian of che Plane and theHour of 7. Again, -

} * A =
‘TD take 86 gr. 5m. out of goer. the common meafure of 6 Hours, -anel
ee there

With this Table thus made you may draw the Hour-lines, and {et up.
_ the Style on either Face of chis or the like Plane, the difference being only -
_ in the placing of the Subftylar, and that is refolved by the fight of the

- cline 85 gr. for the prime Vertical, the one Face of ,it being open to che

Sy, Ge pe a
Rae

j hee
Mick
i

34 «The Deeription of the Hla-lines

TlAn, PolAr. Plalc Jc G| there remains 3 gr. $5 m. for
Gr. Mi\Gr. M,|in. Par.|In. Par.| the. Angle at the Pole between
12186 5/38 a3 or o8l79 21 | che Meridian of the Plane and —

the Hour of 6. To thefe An-
gles fo found, I allow 15 gr.

6334 ; _ i sO 8 fcr every Hour, asin the fecond
roleG) st 36 9 25h Boy Column of this Table. |
pated > Dnaictate. Piha Es Se ae) - Then having the height of

940° °§| * 4255 43,4 a] the Pole above che Plane; and —
826 5| 1 311 3 OS} 2 5) thefe Angles at the Pole, che
Hie ys “b afe RS NN L 5 ©5} Arks of the Plane between the ~
Merid. jSubftyl. | 9 9 3 4 Subftylar and the Hour-circles
138, §5| o of : 3 will be found as in the third
453 53| 2 s}4 18) 3 64) re hee found, will
Me stake heuel efe Arks being found, will
340 55 ; Ah Ge nclele ferve for the drawing of the
ei 55 20|12 77} Hour-lines on either Face of |

I7r 25} 9 10[T8 56116 I) his oy the like Plane |

178 §5it§ 2831 8227 67 - Bes thes hel . h |
be a1 a estoastisate \ et t; By the help of a Thred |

- and Plummet I draw Z Ca |
Vertical Line, ferving both |
for the Meridian of the place, and the Hour of 12. |

2. In this Meridian Line I make choice of a Center inthe upper part _
of the Line, if ichad been the Southern Face of the Plane ; but here in |
C the lower pare of the Line, becaufe we fuppofed itto be the North-
weft Face of the Plane, and the Style muft point upward: and upon this |
Center I defcribe an occule Circle; reprefenting che Declining Vertical |
belonging tothis Plane. ; if -\ alae

3. finda Chord of 38 gr. 23 m. the diftance of the Subftylar from
the Meridian of the Place, and infcribe it into thisCircle, from Z inthe |
Mcridian, unto T coward the left hand, according as the proper Mesidaal

P T falls in the Fundamental Diagram ; andhere I draw the Line CT

7 ferving for the Subftylar,

4. TheSubftylar being drawn, I may infcribe the Chords of the Ark:
of the Plane from the Subftylar, and draw the Hour-lines, and fer up th
Style, as inthe former Plane,

Or the Arksof the Plane from the Subftylar being found as before, w

may draw the Hour-line, upon the Plane otherwife than by Chords: Fo”

having drawn the Hour-lines as in the laft Figure, upon Paper or Balt
boar

Ree Sa ee A Ce a, RP Mt Cy Poe

=

\

board, we thall find che moft part of them, in this and fuch like Planes

_ thac have greater Declination, to fall fo clofe cogether, that they can

_ hardly bedifcerned : wherefore te draw chem atlarge co the belt advan=
tige of the Plane, I leave out che Center, and draw them by Tangents,
_ a31in the Polar Plane, 3 . |

© 3, I confider the length and breadch of the Plane whereon I am to
draw the Hour-lines, which I fuppofe to bea Square whofe Side is 36
Inches, and find chat the little Square A B D E will contain both the Sub-
‘| ga and all thofe Hour-lines which are required in the great Square

| 2, ikaw two parallel Lines, FN,G M, croffing the Subftylar at Righg
_ Anglesin the points F and G,as'they may beft crofs all the Hour-lines,and
_ yet the one be diftant from the other as far asthe Plane will give me leaves

Bee 2 and *

ins a Vertical Declining Planes 2s

duets ;
be a0

Hourelines, the proportion willhold,

{cribe an occult Ark of a Circle..

(a> er Yee Dp. De RS ad De SERS ee ul SS SORT Bes Os ae bee >” a au FP a ae
} FIN 35 Ie ; 7 4 3 x A i soe OEY yay YS SS aS Te eae PONS
\ ri) ee ti ‘ , eta ee oe, q GPM ee © ;
\ . ' 3
bak:

sh

36. The Defeription of the Hour=lines

“ad I find by the fight of the Figure, that if A Behe Side of che leffer 1

Square fhall be 36 Inches, the Line C F willbe abouc 115 Inches, and
the Line C G about roo Inches,’ and therefore FG rg Inches. Again, —
thar che Point F will fallaboue 6 Inches below the upper Horizontal Side!
AB, and about 12 Inchesfrom the nexe Vertical SideBD; for I need ~
not here ftand upon Parts, Sal
3. Becaufe thefecwo parallel Lines are Tangent Lines, in refpect of
Circles drawn upon the Semidiameters CF, CG, and fuch Tangent as
belongs to the Arks of the Plane, being between the Subftylar and the

els the Tangent of 45 gr. :

istothe Tangent of the Ark of the'Plane :
So the Length of the Semidiameter,

to the Length of the Tangent-line.

As for Example: The Ark of the Plane berween the Subftylar and —
the Hour of tis1§ gr. 28%. in che former Table, the Semidiameter _
CF 115 Inches, and the Semidiamerer CG 100 Inches: Wherefore]
extend the Compaffes from the Tangent of 45 gr. unto the Tangent of —
15 gr. 28 ms. the {ame extent will reach from 115 in che Line of Numbers, ©

/ unto 31,82,which thews thelength of the Tangent-line between F in the

Subftylar and che Hour-line of x to be 31 Inches 82 cen, or parts of 100.
Again, che fame extent will reach from 100 unto 27,67; andfuchisthe
length of the leffer Tangent from G to the Hour of 4. —
The like reafon holds for the length of che other Tangents from the
Subftylar co the reft of the Hours, asin the Table; as alfo for che height —
of the Style above thefe Tangent-lines: and fo the Angle of che Style
above the Plane being 3 gr. 6 m, the Height F K will be found to be 6 In-
ches 23 cent. and the Height GL § Inches 42 cent. |
Where the Reader may obferve, thacif the extent from the Tangenc of
45 gr. tothe Tangent of 3 gr. 6 m, orto11ys in the Line of Numbers,
be roo large for his Compafles, he may ufethe Tangent of 5 gr. 43 m. in
ftead of che Tangent of 45 gr. as I noted before. | a |
4.Having found thefe Lengths.and Heights,and fet them down in a Ta-
ble, I come tothe Plane here refembled by the Iefler Square A BDE,
where I begin with an occule Vertical FH, about 12 Inches from the Side
BD, and upon the Center F, about 6 Inches below the Side AB, des

Me |

S. Into if

Pe iP oo he Ba ao BAe le gk sae i Ae Daan 3 fh tie | baa he Ohad aie) = oe Oras Be ae ar
pili Be, ' Fr ‘ C A “2 : A ay
: 7 = 1 4 * 5 : a f ue A

5+ Intothis Ark I firft infcribe a Chord of 38 gr.23 mw. the diftance
of the Subftylar from the Meridian, to make the Angle H F G equal to
the Angle Z CT; fo the Line FG fhall be the Subftylar : and then ano-
‘ther Chord of §t gr. 37 a. the Complement of this Diftance, to make
‘upthe Right Angle GFN ; fo the Line FN {hall be the greater of the

__ 6. I fet off 15 Inchesfrom F unto G toward the Center, and through

G draw the leffer Tangent-line G M, parallel to the former.

7. Thefetwo occult Tamgent-lines being chusdrawn, E look into the
former Table for the Hour of 1, and there findthe Ark of the Plane-be-

- tween the Subftylar and the Hour of t to be 1§ gr, 28 a. and the length

Teffer Tangent-line 27 Inches 67 cent. wherefore I take out 31 Inches
82 parts, and prick chem down inthe greater Tangent from F toN, and
‘then 27 Inches 67 Parts, and prick them down in the leffer Tangent
from G to M, and draw the Line MN for the Hour of 1, which if ic
‘were produced, would crofs che Subftylar F Gin the Center'C, and there
make the Anele FON 1g gr. 28%. The like Reafon holdeth for the
drawing of all che reft of che Hour-lines,

FK may be 6 Inches 23 cent. and the Height G L 5 Inches 42 cent. then:
fhall K L reprefene the Axis of the World, and if it were produced,
_ would crofsthe Subftylar F G in che Center C, and there make the An-
‘gle F CK to be 3¢r. 6 m. and fo be truly placed forcafting of the Sha-
\dew upon the Hour-linesin this Declining Plane,

CHAP. VIII.
To draw the Hour-lines in a (Meridian Inclining Plane.

i :

_A LI thofe Planes wherein the Horizonral Line isthe fame with che
£X% Meridian Line are therefore called Meridian Planes: if they be
right to the Horizon, they are called by che general name of Meridian
‘Planes, without farcher addition, and are defcribed before : if they lean
‘tothe Horizon, chey are chen called Meridian Incliners. :

| Thefe may incline either to the Eaft part of the Horizon, or to the
Weft, and each of them hath two Faces, che upper towards the Zenith,

{
i]

j

the lower towards the Nadir, wherein knowing the Latitude of the

| ieee ‘its a Meridian Inclining Plape; iy
two Tangent-lines before- mentioned. ie

belonging to it inthegreater Tangent-line to be 31 Inches 82 cen?. in the»

Laftly, I fet up the Style right over che Subftylar, fo as the Height.

Place,.

ee

ie 38 The Defcription of the Hour-lines

SR Mia oe Moet URE) SA NEUTER” TSS 2 3 Ste yma RS Can aL a a A
: Goon as * kd TS te . Te - “ait

Place, and the Inclination of the Plane to the Horizon, we ate to com:
fider,’. - i Va
1, The Inclination of che Meridian of the Plane co the Meridian of

the Place. aR

‘a. The Heieht of rhe Poleabove the Plane.

3. The D.ftance of the Subftylar from the Meridian. -

4. The Diftance of each Hour-line from the Subftylar.

And all thefe four are reprefented in the Fundamental Diagram, as in
this Example. a

In our Latitude of st gr. 30m. a Meridian Plane inclinech Eaftward

50 gr. thefe go gr. I number in the Vertical Circle from E unto G, ac

cording to the Inclination of the Plane, and there draw the Ark $GN
reprefenting the Plane propofed. Again, 1 number §0 from Z unto K, |
{o the Point K (being 90 gr. from the Plane at G) fhall be che Pole of
this Plane, and the proper Meridian of this Plane may be fupplied by a
Circle drawn through KandP. This Meridian doth here fall berween |
cheHoursof 4nd 5, and croffing the Planeat Right Angles in the Point |

V, inthe Right Line C V hall be the Subftylar, and the Angle PCV |
the height of the Style above the Plane, and Right Lines drawn from

the Center C to the Interfections of the Hour-circles wich § G N hall be »
the Hour-lines here inquired. The lower Face of the Plane will concaifi,
all the Hour-lines from Sun-rifing unto 11 in the Morning, and the uppet |
Face the Hours from g in the Morning unto Sun-fetting. Then haveLa |
Reétangle Triangle P V N, wherein the Bafe P N is the Height of the Pole —
above the North part of the Horizon, and the Angle PN V the Comple. _
ment of the Inclination to the Horizon: And thefe being known, A |

1. I may find the Angle NP V of Inclination of the two Metidi-

ans: For,

eA: the Cofine of the Latitude,
is to the Sine of 90 gr. :
Sothe Tangent of Inclination to the Horizon,
to the Tangent of Inclination of Meridians.

Extend the Compafies from the Sine of 38 gr. 30%. the Complement:
of the Latitude, unto the Sine of 90 gr. the fame extent will reach from:
the Tangent of 50 gr. 0 m, the Inclination of che Plane to the Horizon,
unto the Tangent of 62 gr. 2§ m.and {uch isthe Inclination of the Me-
sidian of the Plane to che Meridian of the Place; which being refolved:

- inte

/

into Time, doth give about 4 he. and 10 #. from the Meridian, for the
placeof the Subftylar among the Hour-lines. |

;

_ 2, The Height of the Pole above the Plane is here reprefented by the
quantity of the Ark of the proper Meridian P V between the Pole and the
Plane, and may be known by that which we have givenin the former

Triangle PV N. For,

eA the Sine of 90 gre

to the Sine of the Latitude:

So the Co-fine of the Inclination to the Horizon,
to the Sine of the Height of the Pole above the Plane.

Extend che Compaffes from the Sine of 9@ gr. unto 51 gr. 30 m. the
Sine of the Latitude, the fame extent will reach from the Sine of 40 gr.
the Complement of the Inclination of the Plane to the Horizon, unto the
‘Sine of 30gr.12m, Or; | |

_ eAss the Sine of 90 gr.

ta the Co-fine of Inclination of Meridians:

! So the Tangent of the Latitude, 7
| | tothe Tangent of the Height of the Pole abowe the Plane.

_ Extend the Compaffes from the Sine of go gr, unto the Tangent of
| gi gr. 30m, the Latitude of the Place, thé fame excent will reach from
the Sine of 27 gr. 35 m. the Complement of the Inclination of the two
| Meridians , unto the Tangent of 30 gr. 12 m. And fuch 1s P V the
| Height of the Pole above the Plane, and fach muft be the Height of the
Style above the Subftylar. |

3. The Diftance of the Subftylar froma the Meridian is here reprefented
by NV che Ark of the Plane between the two Meridians, and may be
found by that Which we have given at the firft in che former Triangle

PVN. For,

ets the Sine of 90 gr. :
to the Sine of the Inclination to the Horizon :
Sothe Tangent of the Latitude,
to the Tangent of the Swbftylar from the (Meridian.

in a Meridian Inclining Plane. or gest

Extend the Compafies from the Sine of 90 gr. unto the Tangent ef
Sl gr.

ger ier
ee ra
ro 4 i .

oy rs

MED Os: The Defeription of the Hour- lines ee:

FS RT RHE RG Ah A ND mae Beem OM OR AIRES IN. ie. Ngee Prim (Wey eC Nes a ry 3 aS: tea 2) PP ey,
er Oka ok) liad Sie ay DMRS SR MO RRR ne ill FHL PAL 6 6 8 LS etd he aS
~

*

§t gr. 30 m.the Latitude of the Place, the fame extent will reach from’
the Sine of so gr. the Inclination of the Plane to the Horizon, unto the
Tangent of 43 yr. $5 m. -And fuchis the Ark N V, the diftanceof the -

Subftylar from the Meridian. ue’ tg . fd
4. The Diftances of che Hour-lines from the Subftylar are here alfo ©
reprefented by chofe Arks of the Plane which are here intercepted between |
the proper Meridian and the Hour-circles, and may be found by that :
which we have given in the Triangles made by the Plane, with his proper
Meridian and che Hour-circles: For the Angleat V, between the Plane |
-and che proper Meridian, is well known to be a Right Angle, and the Side
PV is che Height of the Pole above the Plane, and the Angles ac the
Pole between the proper Meridian and the Hour-circles are eafily gathered —
. ~ into. a Table. The Angle V PN becween ~

Latitnd 5t 30 V P the proper Meridian of the Plane,and)
Inclinac. JO Oo P N the general Meridian of the Place, bee -
Dit Merid. 62 25 ing 62 gr. 25 m. the Angle between the pro= !

Ay 4
Alt. Scyl. per Meridian and the Circle of the Hour of ©
Dif. Subtt a "2 rt will be 77 gr. 25 m. and the Angle bes -

Ang, Po,JAr. Pla. longing to the Hour of 1,47 gr. 25 a, and —

Hours. (qr Gis fo the reft of the Anglesac the Pole. Then |

»
I | oe a y bo the Sine of the Height of the Pole
2 a 33 [7 a | above the Plane : pa |

3 17 ae i 43 So ibe Tangent of the Angle at the Pole,
a 2 25) 1 13 to the Tangent of the Hosr-line from |
Merid. |S abftyl. the Subftylar. i |
5 27 ¥ < Ag Wherefore I extend the Compaffes from
Bagh els. Bat 8 the Sine of 90 gr. unto the Sine of 30 gre |
8.157 35/38 23 12 m. the Height of the*Pole above the ©
9 172 35158 3 Plane, and | find the fame extent to reach
10 {87 35!85 12179 the Line of Tangents from 77 or. 25 me
re 9) SS nto 66 gr. 4m. forthe diftance belonging —

to the hour of a1 and from che Tangent —
of 62 gr.25 m. to 43 er. §§ ms. for the Hour of 12, as when I found
che diftance of the Subliylar from che Meridian: And fo for che reft of —
she Arks of the Plane between she Subftylar and che Hour-circles, as in
she Table. | Pe. |

Thele

+ eee

(a ina Meridian Inclining Plants, 4
Thefe Arks being thus found, will ferve to draw the Hour-lines on

either Side of chis Plane: But fuppofing it to be the upper fide, _

|. I draw che Horizontal Line CN, ferving for the Meridian and

Hour of 12. ne

| ~ UE Bin

\ “3 at AN 1a
dr < NY ,

a XY ‘
) 2. In this Line I make choice of a Center at C , and thence de- :
feribe an occult: Ark of a Circle reprefenting the Plane propofed. om
|, 3+ Efind a Chord of 43 gr. 55 m. the diftance of the Subftylar from
the Meridian, and infcribe icinto this Circle from N unto A, according,
as I find the proper Meridian P ¥ to fall in the Fundamental Diagramy . vi
and there Ldraw the Line C A ferving for the Subftylar. a

INS Fae

SP Ties ee ea Se ee oe ae
sat te ast Hee eae SPE tae A ee BED Ee
>

Cs Phe Deferiptian Of the Blowin iy
- 4: The Subftylar being drawn, I may infcribe the Chords of the Arks.

_ of the Plane from che Subftylar, and draw the Flour-lines, and fet up the.
Style, asin cheformer Planes. ba bl
ef i

CHAP. IX.

a

Yo draw the Hour-lines in a Polar Declining Plant.

_ PRP Hole Planes wherein a Line may be drawn parallel tothe Axis of the - |
1 World are called Polar Planes, becaufe that Line pointeth unto the:
OT Poles; andthefe Planes are always parallel to fome one of the Hour-cit-
: cles. If chey be parallelto the Hour of 6, they are called Dire Polar
Planes: if to the Hour of 12;they.are called Meridian Planes; and both

thefe are defcribed before : if to any other of the Hour-circles,. they are

~ then called bythename of Polar Declining Planes, becaufe of their ins
clining to the Pole, and declining from the Vertical. ,
Thefe kind of Planes maybe knownin this fore: Firft, confider the-
Tnclinarion of the Plane to the Horizon, which in thefe parts of the World”
muft always be Northward, and more than the Latitude of the Place?
Then find the Declination from the Verticals Thefe two being knowm)’
if the proportion hold, )

3 ds the Sine of goer: a
, to the Co-fine of the Declination « i}
So the Tangent of the Declination,
te the Tangent of the Latitude,
"a itis then a Polar Declining Plane 5 otherwife not.
For example: In our Latitude of §1 gr. 30m. a Plane: is propofed
declining from the Vertical 65 gr. 40 m..and inclining Northward 7Pgrs
51 m.the upper Face being open to the South-eaft, and the lower to the
North-weft. If Inumber thofe 65 gre 40 min the Horizon of the Fun
damental Diagram from EuntoQ, and draw the Line HC Q3 it thal
reprefent the Horizontal Line of the Plane: then croffing it at Righ;
Angles with the Plane BZ D drawn through the Zenith, I aumbe:
qigr. 51m. for the Inclination from Dounto R, and there draw th.
CircleH R Q; this Circle fo drawn thal reprefenc the Plane propofed
and becaufe iralfo pafleth through the Pole, icis cherefore a Polar a.
, Li

‘
> _
SS
iii a in

ae in 4 Polar Dechwing Plane. 4? A
| But for farther trial, I extend che Compaffes from the Sine of 90 gr. to the |
| Sine of 24 gr. 20m. the Complement of the Declination, and I find
the fame Extent co reach from the Tangent of 71 gr. 51 m. the Inclina-
tion propofed, unto the Tangent of 51 gr. 30 m. whichis the true Lati-
tude of the Place ; and therefore it is a Polar Plane. ik

| Again, I number the Inclination of 71 gr. 51m. inthe Circle BZD
from Zunto M, fo this point M will fallar the meeting of B Z D with
‘the Equator, and being gogr. fromthe Plane at R, it fhallbe the Pole
of this Plane; anda Circledrawn through M and P will be the proper

| Meridian of this Plane. This Meridian M P here falling on-the Hour of ~
8, doth give MP Z, the Angle of the Inclination of Meridians, to be
\4, Hours, ox 60 Degrees 5 then croffing the Plane at the point P, it (hews
ichat the Subftylar fhould be CP, and be placed at the Hour of 8, Bue
\becaufe P is-che Pole, and C P the Axis of the World wherein all che
\Hour-circles do méet, and fothere would be no diftinGtion between the
lAxis, the Subftylar, and the Hour-lines, I now fuppofe the Plane in a
parallel to the Circle HRQ., according to the diftance that I would have
beeween the Axisof che Style and the Subftylar, then will che Style be
paral tothe Plane, as appearsin the Fundamental Diagram.

Here then the Style will be parallel to the Plane, and che Hou-lines
parallel one to the other, asin the Meridian and Direct Polar Planes. Yee
that we may better know how te draw the Hour-lines, and where to place
the Style, we are coconfider,

i

. nee a
i. et ;
oe. ple os os eT :
Ped Oe en” ops yO em ae en en CORN eas

u, Lhe Ark of the Plane between the Horizon and the Pole. ae

In a Meridian Plane, the Ark becween the Horizon and the Pole,which —
reprefents the Ark between the Horizon and the Hour-lines, is always
equal tothe Latitude of the Place; ina direct Polar it isan Ark of 90 gre
‘in thefe Declining Polars it is greater than the Laritude, and yet lefs than
gogr. This Arkis herereprefented by PQ. and may be known by ree
folying the Triangle QN P, or PRZ.

As the Sine of 90 gr.
a3 to the Co fine of the Latitude:
_ So the Sine of the Declination, -
| . tothe Co-fine of the Ark between the Horizon and the Pole

_ Extend the Compafies fromthe Sine of go gr, unto the Sine of 38 fre
Frf 2 Dhaere 30 re

ge age ~ f. a.
‘a 6 Ue ea Rata aan 1 ene as i ia Piss a i CES AERC PRN Gann NTR ake Soa CP een a

44 The Defeription of the Hour-lines- ae
30 w, the Complement of the Latitude, the fame extent will reach from
the Sine of 65 gr. 40 m, the Declination propofed, unto the Sine of

34.7. 34 m, whole Complement is 55 gr. 26m, the Ark of the Plane.
required between the Horizon and the Pole, nO

ay Or, 4 the C o- fine of Inclination to the Horizon, |

me to the Sine of 90 gr.

ae | Sothe Co-tangent of the Declination,

ao tothe Tangent of the Ark, between the Horizon and thé Pole..

: And fo extending the Compaffes from the Sine of 18: gr. 9 m, the Come

| plement of che Inclination to the Tangent of 24 gr. 20 m, the Comple.
ment of the Declination, the fame extent doth reach from the Sine of
90 gr.unto the Tangent of 55 gr. 26m. And fuch is QP the Ark of the
Plane between the Horizon and the Pole, the meafure of che AngleQ CB

__-beeween the Horizontal Line and the Subftylar.

ae 2, The Inclination of the Aderidian of the Planetothe Ad&eridian of whe
ah gan Place. |

Fhe Subftylar ina Dire& Polar Plane is always the fame with che

P Hour-line of 12 ; ina Meridian Plane it is the fame with the Hour-line |
: of 6 in thefe Declining Polarsit muft be placed between 12 and 6, ace
E cording to the Inclination of the Meridian of the Plane to the Meridian

of the Place, which ishere reprefented by MP Z,. the Complement of,
the Angle R PZ, and thus known. | Va

|
7 a
As the Sine of 90.2. il
to the Sine of the Latitude’:
So the Tangent of the Declination of the Plane,
- tothe Tangent of the Inclinatton of Meridians.-

Extend the Compaffes from the Sine of go gr. to the Sine of 5 grs
30 m.the Latitude of the Place,the fame extent will reach from theTangent
of 65 gr. 40m. the Declination propofed,' unto the Tangent of 60 gr.
| and fuch is the, Angle of Inclination between the Meridian of the. Place
iM. and the proper Meridian of the Plane, which refolved into Time, doth >
: make four Hours; and fo the Subftylar muft here be placed upon che Hour
of 8 inthe Morning. wiht : ie |
is

a“

Oe ee

Wa

This Angle being known, the reft of the Aneles at the Pole are eafily

gathered: For if che Hour of 12 be 60 gr. diftan: from the Meridian of
the Plane, the Hour of « willbe 75 gr. and the Hour of 11 will be 45 gre

diftant, andthe reft of che Hours, asin che Table following. Then co-

ming to the Plane, :

__ 1,.1 draw-an occult Horizontal Line HQ, wherein. I makechoice of a
Cenerat H, and defcribean occult Circle for the Horizon of the Plane.

2» I finda Chord of §§ gr. 26..and infcribe it into this Circle from.
Q ynto B, according to the fituation of the Plane; fothe Line HB thall

‘bethe Meridian of the Plane, and therefore the Subftylar; and the Line
AC, crofling it at Right Angles, thall be che Equator. /

h
|
)
it | F
- CONGO BOLANG UNOS TAIT A NTGNG TUG SO RAINED O20 WETS ane a0 UAGRARAARSI AGA» Wore » bis
es A 5 : X

) ‘
| ’ : Ix

__ 3-Tconfider the length of the Plane, and how many Hours 1 am to

_ draw upon it, that:fo I may proportion the Height of the Style’; and I

find by the Fundamental Diagram, and the former Table, that ic will
, i contain

I \

|
|

ina Polar Declining Plane.’ 4S

il

46 The Defcription of the Hour-lines

contain all che Hours from Sun-rifing until ic be paft x after Noon: and _

therefore the Meridian of the Plane falling on the Hour of 8 in the morn=

ing, there will be four Hours on the one fide, and’five oa the other fide
of theSubftylar. But in all Polar Planes the height of the Styleabove the

Subftylar muft be equal to the diftanceof the third Hour from the Subfty-
far, or about F of the fourch Hour, or little more than + of thefifth Hour,
aad thereupon I allow the height of this Scyle to be equal to C B, whica
youmay fuppofe to be ro Iuches. ca.

4+ Becaufe the Equator-A Cis a Tangent-line;in refpe& of the Radius :
BC, and the pares thereof are fuch-as belong to the Angles between‘the -
Meridian of the'Plane and the Hour-lines, which Angles are fet down in

the Table following, Imay find the length of each feveral ‘Tangenc in |

this manner.

As the Tangent of 45 gr.
is tothe Tangent of the Hour: -
Sethe Parts of the Radius,

tothe Parts of the Tangent-line,
‘|Latitude sl 30 “The Angele AB C between the Meridian

Declinat, 65 40

Inclingt, -971 ~5t

Diff. Merid, 60 0
\Dift, Subft. 55 20 |

_ |Angs Po.(= Fang.

*|Gr. Mtn. Far,

reach from fo in the Line of Numbers, unto.

~ seafon holds for the reft of the Hours.

| Merid. |Subityl. ., -Thefe Lengths being chus-found and {et
9 |FS oo] 2 68) ;

of thePlane and theHour of 43, the Merie _
, dian-of the Place, is Go gr. in che former.

Table, and the Radius BC is fuppofed co.
be 20 Inches whereupon I extend the
Compaffes from the Tangent of 45 gr. unto
the Tangent of 60 gr. the {ame extent will |

“down in the Table, I take out 17 Inches
+32 cent, and prick them in the Equator from
C unto A for the Hour of 12, and 37 In-

q

17.32, which fhews the length of the Tane
gent AC, betweea the Subftylar and the —
Hour of 12, to be 17, 32cent. The like |

0 OO17 32) ches.32 cent. and prick chem down for the ©
T 475 0037 = =32% Hour of r: And fo the reft of the Hour-
2 490 oollnfinie. | 3

points.

6. This done ‘ if I draw Right Lines

through |

|

) 3. The Ark of the Plane between the Horizon and the Meridian- :

i 4 isi wtih
| ie ine Declining Inclining Planes - ee,
through each of thefe Points, croffing the Equator at Right Angles, they
“fhall be the Hout: lines required: And if T fet the Style over. che Subfty-
Tar, fo as the edge of it may be parallel ro the Plane, and the height of ic
be roInches, equal to the former Radius B C, ic fhall reprefent the Axis

| of the World, and be truly placed for cafting of the Shadow upon the
|" Hour-lines in chis Declining Polar Plane. |

To draw the Honr-lines ina Declining Inclining Plants

| [' a Plane thall decline from the prime Vertical, and incline to the
| Horizon, and yet not lie even.with the Poles of the World, itis then
called a Declining Inclining Plane. : .
OF thefechere are feveral forts 5. for the Inclination being Northward,

| the Plane may fall beeween the Horizon and the Pole, as the Circe BMD

BED; orthe Inclination maybe Southward, and fo be reprefented by
BED: it may alo fall either below the Interfe@ion of the Meridian and

| the Equator, or above it; and each of thefe hath two Faces, the upper -
| toward the Zenith, and the lower toward the Nadir; wherein having
| the Latitude of the Place, with the Declination and Inclination of the..

| Plane, we are further co confider,
4, The Ark of the Meridian between the Pole and the Plane.
2. The Inclination of the Plane to the Meridian.

,

4, The Angle of Inclination between both Meridians.
_. g« TheHeight of the Pole above the Plane.
6+ The diftance of theSubftylar from the Meridian.
9. Thediftances of each Hour-line fron the Subftylar. :
And all thefe feven may be reprefented in the Fundamental Diagram,
asin chis Example.
— Inour Latitude of 5 gr. 30m. a Plane is propofed declining from the
Vertical 24 gr. 20 m, and inclining Norchward 36 gr. the upper Face
"lying open to the South-weft; the lower to the Norih-eatt. If T number
thefe 24 pr. 207, in the Horizon from Eto B, and there draw the Line
BCD, 1c thall reprefent the Horizontal Line of the Plane : Then crof-
fing it at Right Angles with the Plane HZ Q drawn through the Zenith,
' | | I

jn the Fundamental Diagram; or between the Zenith and the Pole, as.

ate
us
e
:

ut

4s The Defeription of the Hour-lines th
I number 36 gr. for che Inclination from Q_unto M, and there draw —
the Circle BMD, croffing the Meridian in the Point 4; this Circle fo —
drawn hall reprefencthe Plane propofed :-and becaufe ic doth not pafs
through che Pole, is therefore no Polar, but an ordinary Declining In-
clining Plane. iN | ! -

1. The Ark of the Meridian of che Place between the Pole and the
Plane is here xeprefented by P 2, and may be found by refolving che Tri-
angle D N «,. wherein the Angle at N is known to be aRight Angle, che
Angle at D is the Angle of Inclination, the Side DN the Complemenec —
of the Declination; which being known, :

As the Sine of 90 gr.
‘to the Co-fine of Declination’: =
So the Tangent of Inclination to the Horizon, | na
to the Tangent of the Ark, of the (Meridian between the Horie
zon ang the Plane, , 7 | y |

Extend ‘the Compaffes from the Sine of 90 gr. unto the Sine of 65 gre
40m. the Complement of the Declination, the fame extent will reach
from the Tangent of 36 gre the Inclination propofed, uncothe Tangent’
of 33 gr. 30 m. and fuch is the Ark of the Meridian N 4 beeween the
Horizon and the Plane. This Ark Na being compared wich the Ark _
NP, which isthe Elevation of the Pole above the Horizon, andis here |
fuppofed tobe 51 gr.30 m. the difference N 4 cometh to 18 gr. and fuch
‘is the Ark of the Meridian required between the Pole and the Plane.

2. ‘The Inclination of: the Plane to the Meridian is here reprefented by
the Angle Naz D, and may be found by chat which we have given inthe -
‘former Triangle DN.4. For,

As the Sine of 90 gr. |
to the Sine of the Declisation from the Vertical :

So the Sine of /nelination to the Horizon,
to the Co-fine of Inclination of the Plane to the (Meridian.

Qo —

Extend the’ Compafies from theSine of 90 gr. unto the Sine of 24 gn,
20.%.the Declination of the Plane, the fame extent will reach from the
Sine of 36 gr. the Inclination given, ‘unto the Co-fine of 76 gr. and fuch”
is N a Dthe Angle of Inclination between che Plane Dz and N athe Me-

ridian of the Place. Or, | a)
As

a a ™
_ =
x

Pao ee ee ‘ an

“4s the Sine of the Ark of the Meridian between the Horizon and the
Plane, ik £
is to the Sine of 90 er.
So the Co-tangent of the Declination,
to the Tangent of Inclination of the Plane to the Meridian.

7 Extend che Compafles from the Sine of 33 gr. 30». the Ark of the

_ Meridian between the Horizon and che Plane, unto theSine of go gr. the
_ fame extent will reach from the Tangent of 65 gr. 40 9. the Complement
_ of the Declination, unto the Tangent of 677. and {uch isthe Inclination
of the Plane tothe Meridian, the fame as before. aire F

3. The Ark of the Plane between the Horizon and the Meridian is here
reprefenred by D 4, and may alfo_be found by chat which we have given
in the former Triangle D N z.

eAss the Co-fine of Inclination to the Horizon,
is to the Sine of 9@ gr.
So the Co-tangent of the Declination,
to the Tangent of the Ark of the Plane from the Hurizon to the
Meridian. |

Extend the Compafies from the Sine of 54 gr. the Complement of the

| Inclination of the Plane to the Horizon, unto the Sine of 9® gr. the
| fame extent will reach from che Tangent of 65 gr. 40 m. the Com-
' plement of the Declination, unto the Tangent of 69 gr. 54m And

tuchis Dia the Ark of the Plane between the Horizon and the Meridian

of the Place. : .

4. The Inclination of Meridians is here reprefented by the Angle 2b P.’

~ For having drawn the proper Meridian bP k, or let down a Perpendicular
Pb fromthe Pole unto the Plane, this Perpendicular fhall be the Meridian

of the Plane, and we fhall have another Triange 4b P, wherein the Angle

at bisa Righe Angle becaufe of the Perpendicular,the Angle at 2 is the In-

_ clination of the Plane to the Meridian of the Plage, and che Side P a is the

Ark of the Meridian between the Pole and the Plane; which being known,

As the (0-fine of the Ark of the Miridian letween the Pole and the
Plane,

. is tothe Sine of 90¢r. 3

Gge So

jn a Declining Inclining Planes gg

RN, eA 7 RT AW Py) A ea DA a 2 ON ee | ‘a ee (a0 ee
\ ‘ Set ) We : Shr wre x Lie a af 5 Tie hi Aa ie de ab VOU Gals he
¥ mt Oy . Mey, M

5 ee ht ie ys
? Ne ‘ fee We s , . 7 es &
,

$0 The Defeription of the Hour-lines

So the Co tangent of the Inclination of the Plane to the Meridian,

go the Tangent of Inclination of the Meridian of the Plane tothe —
«Meridian of the Place, that is, the Angle at the Pole between
thetwo:Meridiante ie ae

+,
mil! |

Extend the Compaffes from the Sine of 72 gr. the Complement of the —
‘Ark P 4 between the Pole and the Plane, unto the Sineof 90 gr. the fame
excent will reach from the Tangent of 14 gr. the Complement of the In-
clination of the Plane to the Meridian, unto che Tangent of 14 gr. 41
-And {ach is the Angle a Pb of Inclination becween the Meridian of the
_ Place and the proper Meridian of the Planes which refolved into Time, _
doth make about 59 m. and fo the Subftylar muft here be placed near the
Hour of 1 after Noon.

“g. The Height of the Pole above the Plane is here reprefented by Pb
the Ark of the proper Meridian between the Pole and the Plane, and may.
be found by that which we have given in the Triangle 4b P. For,

As the Sine of 90 gre ‘a
to the Sine of the Ark of the Meridian of the Place between the
Pole and the Plane: ‘

Sobhe Sine of Inclination of the Plane to the Meridsan,
to the Sine of the Height of the Pole above the Plane. ie
Extend the Compaffes from. the Sine of 90 gr. unto the Sine of 18 ord
the Ark Pw of the Meridian of the Place from the Pole to the Plane, the
fame extent will reach from the Sine of 64 P the Inclination of the Plang _
to the Meridian.of thePlace, unto the Sine of 17 gr.26m Ory, :
Ass the Sine of 90 ge q
to the Co-fine of Inclination of Aderidians : e |
So. the Tangent of the Ark of the Meridian. of the Place between the

Pole and the Planes | , ‘|

tothe Tangent of the Height of the Pole above the Plane,

Extend the Compafies from: che Sine of 90 gr. unto the Sime of 75 gre
g.m. the Complement of a Pb the Inclinacion of che cwo Meridians, —
the {ame extent will reach from. the Tangent of 18.gr. the Ark P a of the.
general Meridian between the Pole and the Plane, unto the Tangent |
of 17.gr. 26 m. And fuch is Pé the Height of the Pole Poe the -

| Plane >:

tied

; Fie » ed ATMA tas Vi ae, ee Oe eye ‘ oF ' ’ 7
‘ wy : 7 i Sar hes
64 a ee Ny § / at

ae ap ti

> - 4aa Declining Inchining Planes Be

Plane; and fuch muft be the Height of the Style above the Sub(tylar.

| gogreunto the Sine of 14 gr. the Comple- | Declinac. 24. 20
meat of bP che Inclination of the Plane jInclin.N. . 26 0
to the Meridian, che fame extent will reach [Al-Mena. 6g ca
| from thé Tangent of 28 gr. the Ark of the Dif, Metid a ve

Plane, unto the Tangent of 4 gr. 30m. /Nift,s

And fuch isthé Ark of the Plane between Subic A130
the rwo-Meridians; and fuch mult be the _ | Hours. | bere sus
|, Diftance from the Hour of 12 to the Subs {_ Gre M\Gr. M.

- thofe Arks of the Plane which are intercepr-

“Hour-circles: For in théfe Triangles, the
| Angle at bbecween the Plane and the pro-
per Meridian is a Right Angle, the Side P é
qs thé Height of the Pole above the Plane,
and then the Angles at the Pole berween the
_ proper Meridian and the Hour-circles being

gathered intoa Table,

6. This Diftance of the Subfiylar from the Meridian of the Place is
here reprefented by 2b the Ark of the Plane between the two Meridians,
and may be found by that which we had given at the firft in the former

Triangle bP. For, - - |

'As the Sine of 90 gt.
to the Co-fine of the Inclination of the Plane tothe Meridian :

Sothe Tangent of the Ark of the Miridian of the Place between the

Pole and the Plane,
nnto the Tangent of the Sabjtylar from the Meridian of the Places

Extend the Compafles from the Sine Of Laricude 51 30

general Meridian between the Pole and the j},, Styl 17 26

LA ‘

ftylar. f vA |
9. The Diftances of the Hottr-lines from 8174 4147 35
the Subftylar are here alfo reprefenred by 9 9 41/27

ed becween the proper Meridian and the Ir 29 41) 9. 41

Oo Ig| 0

_ As the Sine of 9° gr.
"to the Sine of the Heightof the Poteabove the Plane-s
Sothe Tangent of the Angle at the Pole,
£0 the Tangent of the Hoar-line from the Subffylar.

Ggg2 Extend

us ( 89 41/58 §7/|

52 ) The Defcription of the Hour-lines

reach from the Tangents of 14 gr. 41 m, the Angle ac che Pole belong-

ing co the Hour of 12, unto the Tangent of 4¢r. 30m. for the Ark of che s
Plane between the Sub{tylar and the Hour of 125 and from the Tangent

of 29 ¢r. 41 m. unto the Tangent of g.gr.41 m. for the Hour of 11;

And {o for the reft of the Arks of the Plane becween the Subftylar and

the Hourelines,. as in the former Table. |

Thefe Arks being thusfound, will ferve for che drawing of the Hour-
- Lineson either fide of che Plane: but fuppofing it to be the upper fide, I.
confider how the Lines do fall inthe Fundamental Diagram, and ac-_ |

cordingly,

Ow —

r L SS

mi i igs wr an TE We eS po laacas

i

ta [draw an occult Horizontal Line D D, wherein I: make choice ‘ ;
P \ ' C

i |

Extend the Compafies from the Sine of 90 gr. unto the Sine of 17 gre |
26 m. the Height of the Pole above the Plane, the fame extenc will —

-

<a
+ |
iS 9s

Me Re ee et wk Ca
MA Declining meclining Planes 5 ?

| the Cencer,G, and thence draw an occult Circle for the Horizon of the
Plane.

Horizon and the Meridian, and defcribe into this Circle from Dunto
| a@y andthere draw che Line’ Ca for the Hour of 12, |

3. I finda Chord of 4 gr. 30m. the Ark of the Plane between the

two Meridians, and inferibe it into this Circle from aunto 6, and there

|
> ||

~~ clination, wit th the Angle of Inclination to
the Horizon at D, and the Right Angle at
_N, thefe former CiReR will give Nd, the

ee

BETO: eT I
Ark of the Meridian between the Horizon Mes) subtiyl
“and the Plane, tobe 74 gr. 20 m, and there- 2 14 43/1 40
' fore P D, the "Ark of the Meridian between 3 |t9 4317 16
“the Pole and the Plane, willbe 22 gr.50 m, 4 (34. 43|t3- 50
the Angle D dN of the Inclination of the 5 |49 43122 46
Plane to the Meridian will be found to be 6 54 43137 0
- 66 gr.29m. and Dd the Ark of the Plane 7 Io 43162 §

i draw the Line Cd for the Subftylar.

4. The Subftylar being diawn, I may inferibe che Chords of the Arks
| of the Plane from che Subftylar, and draw the Hour-lines, and fer up che
Style, asin the former Plane,

of Second Example of a Plane falling between the Pole and the Zenith.

~~ Tn like manner if in our Laticude a Plane
be propofed declining from the Vertical
24 gr. 20 m,.as before, but inclining. to the
Horizon 75 gr. 40 m, Northward, the up-
per Face being open co the South-wet, the
: lower to the North eaft, chis Plane fhall be
| Rere-reprefenced by the Circle BF D, crof-
fing the Meridian in the point d, between
“the Pole and the Zenith, and che proper Me-
‘ridian of this Plane, by the Perpendicular
Ark Pe. i
- Then in this Triangle DN d knowing
the Side DN, the Complement of the De-

Lacatude sl 30
Declinarc, 24°20 +
{nclinar. TS Os
Alt.Merid.. 83 36 —
Diff. Merid. 25 19
O.ft. Subft.-- 9 32
Alt. Styf, 20. 50 |
uf An. Po,| Ar. Ar. Pla.

ae Gr. M. M.|Gr. M.

“berween the Horizon and the. Meidianiisc ities kel

rig £7.36 ed

| 2. [finda Chord of 69 gr. $4 m. the Ark of the Plane between the

{tance of the Subftylar from the Meridian about 9"¢r. 32 m. ae

_an, the Subftylar, andthe Hourclines, and fet up the Style, as in the »
other Plaries. | a |

bn enim! Horizon at By at the Right Angle ar $, we -

SYS OE oe eT es Aken RG Se

CUM A a a als RM ebb

¢ ¥ ~ ay ay the iy ab”,
+ , sa ;

4 ——=—=«s«Te Defeription ofthe Hour-Tines

Again, inthe Triangle Ped, knowing the Side Pd, the Ark of there:
Meridian between the Pole and che Plane, with the Angle of Inclination |
to the Meridian at d, and the Right Angle ate, the Angle d Peof theIn- .
clinition of the two Meridians will be found to be 25 gr, 17m andPe -

the Height of the Pole above the Plane tobe 20 gr. 50 m. and de the dis -

° a

\

Laftly, having found the Height of the Pole above the Plane, and ga= -
thered che Angles at the Pole, the Arks of the Plane from che Subftylar
to the Hour-lines will be as in the Tables m

This done, if weconfider how the Lines: do fall in the Fundamental |
Diagram, we may there fee how the North Pole is elevated above the
lower Face, and the Souch Pole above the upper Face of the Plane, and |
accordingly make choice of a Center, draw the Horizontal, the Meridie

}
j

Declinar. ~ 24 20 the Southward.
Inclinac. Irie) 20" |
Dit Merid. 13 29 If in our Latitude a Plane were propofed |
IDS Sub. t2- 8 declining from the Vertical 24 gr, 20 m, a5 |
Alc. Styl. 64 0 before, but inclining to the Horizon 14 fr.
Alc. Merid.. 66 120 _20 m. Southward, the upper Face being open |
RGR TAG Dia) (tte North-eaft, the lower to the Soath-
Hours. |e ar Gia}. Weft, this Plane thall be there reprefented
eis pg \Gr. -CMh |G ,| bythe Circle BK D, croffing the Meridian —

Latitude st 30 4 third Example of a Plane inclining toll |
3 |

¢ i 3347 2 : inthe point f, between the Equator and che
fees z st 3° Horizon, and the proper Meridian of this
3 146 33 " 3*/ Plane, by the perpendicular Ark P £5 ed
? es 3 j ) 6 _ down from the Pole to the Plane, near the
of . 3 4 41 -Hourof x1, at the North pare of the Ho-

- eae rizon, as may partly appear by cheneareft
sb Veber extent of the Compafles,if the Circle BKD

20 [nz ) 20/02 were drawn round, and the.two Letters f _

1 }28 713 37 and g {upplied. mae H,

2 “a L a 7 : _ Then in the Triangle BS f, knowing the |
3°15 ff H rm Side BS the Complement of tlie Declina-

| : ae au Bah ae tion, with che Angle of Inclination to the |

may

eee ee es ee a inom Vl? weer. yes Xue ae ae ,
es ae St a iB I Sale ae gic a a
P . * 4 * 5 ‘

ee in a Declining Inclining Plane. Fy"
oe find S f the Ark of the Meridian between the Horizon and the Plane’
obe 13gr.6m. And therefore P f, the Ark of the Meridian between
the Pole and the Plane to the Southward 115 gr.24m.. but 64 gr. 36 mm:
to the Northward, the Angle B f S, or Df N of the Inclination of the
Plane tothe Meridian will be found 84 gr. 9m. and Bf the Ark of the
Plane between the Horizon and the Meridian 66 gr.20m. 2
_ Again, inthe Triangle P ¢ f, knowing the Side P f the Ark of the
Meridian between the Pole and the Plane, with the Angle of Inclinati-
on to the Meridian at f, and the Right Angle at g, the Angle f Pg of
the Inclination of the rwo Meridians will be found to be 13 gr. 27 77.
Boe P g the Height of the Pole above the Plane, abour 64 gr. and f g the
D ftance of the Subftylar from the Meridian #2 gr. 8 m.
Having found the Height of the Pole above the Plane, and gathered’
the Anglesat the Pole, che Arksof the Plane from the Subftylar to the.
Hour-lines will be found as in the Table.
This done, if. we confider how the Lines do fallin the Fundamen-
al Diagram, we may there fee how the North Pole is elevated above the
‘apper Face, and the South Pole above the lower Face of this Plane; and¢
accordingly make choice of the Center, draw the Horizontal, the Meri- _
dian, the Subftylar, and the Hour-lines, and fet up the Style, as in the for

}

mer Planesy..* |
| ‘CHAP. XL :

| Todefcribe the Tropicks and other Circles of Declination in aw:
| Yern* - — Eguinettial. Plane.

\Ulch Circlesagare parallel to che Equino€tial, and yee fall: within the

S Tropicks, may be defcribed on any: Plane by help of. chefe Lines of

sroportion, butafter a different manner, according as the Style fhall be:

sither perpendicularor parallel co the Plane, or cutthe Plane with Ob-

Jigue Angles. me! :

| Inan Equinodial Plane, where the Style is perpendicular. to the: -

Plane, the Tropicks and other Circles of Declination willbe perfect Cir-

les: Wherefore confider the length of the Style in Inchesand parts, and:

the Declination of the Circle*which you intend to defcribe in Degrees. fe th
and Minutes, the proportion will hold, te A | ‘A
, ‘ ' : | MG: ii ‘
| | 4
hoa | | ny

i)

_ be) i
|
/ . |
/ / R 5 ii
f " : . oe
} ‘

dae tl Vk es A GE Pe a t Ries > 5 Vie if’ ee pe
= SINE a Sei AG wr) tis ae
‘ * mY, . : * ed

es the Tangent of 45 gr.
to the Length of the Style:
Sothe Co-tangent of the Parallel,
to the Semidiameter of hu Circle. |

Suppofe the Length of theStyle above the Plane tobe 10 Inches, and.
that.ic were required to find the Semidiameter of the Tropick, whofe
Declination is known to be 23 gr. 30%. Extend the Compafies from |
the Tangent of 45 gr. unto the Tangent of 66 gr. 30 a. the fame extent

Ty

ai

wma
wanpatts

6;

S ie
; Ne. Y a
° > rc t
: Ce, ‘
4, |
rm a 4
é Aer . |

SG aray ste md

(TES. \

will reach in the Line of Numbers from 10 unto 23, which thews the
Semidiameter of the Tropick to be 23 Inches. Soif the Declination be
20 gr. the Semidiamer will be 27 Inches 47 cent. if 15 gr. then 32.325.
if xogr. then 96.713 if 5 gr. then 114. 305: and fo inthereft, ©
Ox if it were required to proportion the Style to the Plane,
AAs

aad Circles of Declination, 59

t ibe

| As the Tangent of 45 gr.
to the Tangent of the Declination:
i «So the Semidiameter of the Plane,
sto the Length of the Style.

Asif the Semidiameter of the greateft Parallel upon the Plane were
> but fix Inches, ‘and that Parallel fhould be the fifth Degree of Declinati-
ons extend the Compafles from the Tangent of 4§ gr. unto the Tangent
of 5 gr. the fame extent will reach in the Line of Numbers from 6, 0
unto about o. 53, which fhews that the length of the Style muft be 53
parts of an Inch divided into 100: Then the length of the Style being
_ known, the Semidiameter of the other Circles will be found as before.

| Ibegin here wich the fifth Parallel, and thence preceed unto the Tro-
pick, becaufe the Shadow of the reft near the. Equinodtial would be
over-long, and the Equinodtial it felf cannot be defcribed. The Paral-
Tels of North Declination are tc be fet on the North Face, and the Paral-
els of South Declination on the South Face of the Plane. Neither need
thefe Parallels to be drawn in full Circles, but onely to the Horizontal
Line, which fhall be defcribed in Chap. 18. |

| Having by thefe means fet up the Style to irs true Height, and drawn
‘te Cie of Declination, if we fhall place the Plane fo as it fhall make
at Angle with che Horizon equal to the Complement of the Latitude,
and then turn it uncil che top of the Style caftthe Shadow upon the Pa-
allel of Declination belonging to the Time, the Meridian of the Plane
will thew. the Meridian of the Place, and the Shadow of the Style che
Hour of the day, without che help of a Magnetical Needle.

[3 CHAP. XII.

To defcribe the Tropicks and other Circles of Declination in a
he Polar Plane.
JN all Polar Planes, whether they be parallel tothe Meridian, or tothe
| . Circles of the Hour of 6, or otherwife declining, the Equinodtial
ae Right Line, buc the Tropicks and other Circles of Declination
will be Se&tions Hyperbolical, and be thus defcribed.
Confider the length of the Scyle, the Declination of the Parallel, sit |
IaH Hhh the

|
|
4

| es renee | qe
58 The Deferiptios of the Tropicks |
the Angle at che Pole between che Subftylar and the Hour-line, whereon |

you mean to defcribe the Parallel. “ |
Jf you would find where the Parallels do erofs the Subftylar,

eAs the Tangent of 45 ers
to the Tangent of Declination :
So % the Length of the Style, |
to the Diftance of the Parallel from the Equinoktial..

‘Asin the Example of che Polar Plane, where che length of che Style |
BC was found to be r Inch 61 cent, If you defire to know the diftance.
between the Equinoétial and the Tropick upon the Subftylar Line, ex-
rend the Compaffes from the Tangent of 45 gr. unto the Tangent of
23.gr. 30m. the fame extent will reach in the Line of Numbers from
2.61 unto 0. 703 and therefore the diftance required is -7o parts: of an”
Inch divided into 100. The like reafon holdeth for all other Paratlels
of Declination croffing the Subftylar. i

But if you would find where the Parallels do crofs any other of the
Hour-lines, firft find the diftance between the Axis of the Style and the
Hour-line, then the diftance between the Equinoétial and the Parallel:
Both thefe may be reprefentedin thismanner, — _ ie |

On the Center B, and any Semidiameter BD, defcribe an occule Ark
of aCircle, and therein infcribe a Chord of 23 gr. 30 m. from D-unto
T, with fuch other intermediate Declinations as you intend to deforibe
onthe Plane; fo the Line B D fhall be the Equator, and B T the .
| Pick

° and Civeles of Declinations Ky
pick, andthe other.intermediate Lines the |
~ Lines of Declination.

That done, ‘confider your Plane, which
for example may be either the Meridian or
the Declining Polar Plane; wherein ha-
“ying drawn both the Equator and the
~ Hour-lines. as before, firft take out the bf fm.
- Height of rhe Style,and prick that down in oe:

this Equator from Bunto C; then taking
_outall che Diftances between B the top of
the Style, and the feveral Points wherein
the Hour-lines do crofs the Equator, tranf-
fer them into this Equator B D from the
Center B, and at the terms of thefe Di-
“ ftances exe& Lines perpendicular co the
Equator, croffing the Lines ef Declination,
and note them with che Number of the
Hour from whence they were taken: fo
“thefe Perpendiculars fhall reprefent thofe
Hour-lines, and che feveral Diftances be-
| tween the Equator and the Lines of De-
dination hall. give the Like Diftances be-
‘tween the Equator and the Parallels of
Declination upon your Plane, Upon this
‘ground it followeth,

j

| mt To find the diftance between the Axis and the Hour-lines.

I
1

As the Co-fine of the Hour from the Subftylar,

| isto the Sine of 90 gr.

| Sothe length of the Style, |

| to the difance between the eAxis and the Hosr-line.

-~ As if in the former Example of the Meridian Plane, where BC che

height of the Style is {uppofed to be to Inches, it were required to find

the diftance between B tothe top cf the Style, and the point wherein

the Hour of rt in the Morning doth crofs the Equator, which ishere re-

-prefented by Bs, ‘becaufe it is the fifth Hour from the Subftylar, whofe

Angle at the Pole is 75 er. Extend my naa from the Sine ot 35 sre |
ih 2 the

the Complement of the fifth Hour from the Subftylar, unto the Sine of |
go gr. the {ame extent will reach from 10,00 in the Line of Numbers, |
unto 38.645 and therefore the diftance B §, between the Axis and the |
Hour-line, is 38 Inchesand 64 cent. and may be called the Secant of the
Hour. Theninche Rectangle B 5 T, having the Side B 5, and the An- |
gle of Declination at B, |

To find the diftance betweenthe Equinottial and the Parallel.

eAs the Tangent of 45 gr.
to the Taagent of the Declination,

So the diftance between the Axis and the Howr-line,
tothe diftance between the Equinottial and the Parallel.

Extend the Compaffes from the Tangent of 4g gr. unto the Tangent
of 23 gr. 30%. the Declination of the Tropick , fo the fame extent ae 4
. reach

-

and Circles of Declination: aS:

_ reach in the Line of Numbers from 38.64 thediftance between the Axis

and che fifth Hour-line, unte 16.80; and therefore the diftance is §6

Inchesand 80 cent, ~The like reafon holdeth for all the ref{t, which may

be gathered, and fet down in fuch a Table as thisewhich followeth,
Wherein I have fet down the Diftances for feveral Declinations, for

IT gr. 30 m. tor 16.gr. $5 ms, for 20 gr. 13 ms. for 20 gr,4tm, and for
_ the Declination of the Tropick 23 gr. 30 m. which may be applied to the
_ like Declinationsin all Meridian and dire&t Polar Planes, «

= SS

ws SS oe
a

As in the former Example of the Polar Plane, where B.C the heighe
of the Style is found tobe 1 Inch Ox cent. if ic were required to find the
diftance between B the top of the Style, and the Points wherein the

Hour-lines of 7 in the Morning or g Afternoon do crofs the Equator
(which diftances Icalled the Secants of thofe Hours) either you may ex-

tend the Compaffes from the Sine of 15 &r. the Complements of the

_ Hour from the Subftylaty unto the Sine of 90 gr. fo the fame extent will
| reachsin the Line of Numbers from 1, 61 the length of the Style, unto

6. 25, according to the former Canon, Or elfe you may make ufe of the
following Table, extending the Compaffes in the Line of Numbers from

| To. co the length of the Style in the Table, unto 1, 61 the length of the
| Style belonging to your Planes fo the fame extent (hall reach from 38.64,

the Secantin the Table, unto 6, 21, and fuch is your Secant required,
the diftance between the cop of the Style and the point of Enterfection,
wherein the fifth Hour-line from the Subftylar doth crofs the Equator.
Again, the fame extent will reach from 16. 80 the diftance in the
Table belonging to the fifth Hour-line between the Equator and the Pa-
rallel of 13 gr. 30m. declination, unto 2.70 forthe like diftance upon.
your Plane ; and fo for the reft, which may be gathered, and fet down:
ina Table, : ah eu be | |

Hourss

62 The Defeription of she Tropicks

-~

_naermeneng peace mth A A A LESS AL A ALE A TES,
An.Po.| Tang. |Secant.[rx 30|16 §§|20 12/21 41] 2 3

eecreneroome | @emmenniiee | @eeemeemey | eee,

mo) |

Ree | | ee ff i OO oe eee
ay | ee

OS, eee (ee a le A
Sentuettiearanmeaenandl
—_

2 O§$| 3 97) 3 71} 4 O1
2 07] 3 10) 3°75] 4 OF
2 10, 3 15} 3 81] 4 12

32/10
~ {18 45} 3 39]TO : 2x51 13 211 3 oof 4.20

a |
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O
Loe |
ARA
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20] 3 290) 4 Go} 4+ 30
3 39) 4 24] 4 45)
34) 3. $1 4. 60

=e

CONT A

NWN N
we
an
ri pe ff
O
ww Ww

———Sa ee —
== ———————

NWN WN
“XN
O
p.
Oo
WA
fh
co
Ve)

3145 O[1O 0014 14

———

| 4t 15| 8 77/13 30

ey tes © eee te actinides |
SE its

48 45/11 40/15 17
52 30/13 03116 43
56 1§|14 97/18 09
4.|60 OjI7 32/2@ 00] 4 ©7| 6 08] 7 36] 7 95
~ (63 45120 28/22 61| 4 60| 6 88] 8 32] 9 00
67 30]24 14/26 13] 5 311 7 95] 9 O1/10 3911 36
71 1§|29 463% 11] © 331 9 47\11 45|12 37}13 §3
$|75_0137 32138 64| 7 86|11 74/14 20/15 36)10 80
78 45|50 27|51 26|10 43|15 60]18 8o]60 38/22 28
82 30|75 96176 61|1§ §3|23 32/28 10130 49/33 31)
86 19192 §7|152 9O]3I 19146 44/56 26|60 81166 48
6l90 of[nfin. |Infin, |Infin. |Infin. {Infin. [Infin. | Infin.

WN Ww W
We
py
oh
re)
O
aH
O
f

441 3 06} 4 ; fs

The Tangents and Secants in the third and fourth Columns of this Ta-
ble are-taken out of the Tables of the Natural Tangents and Secants, ac-

cording to the Degrees and Minutes that are in the fecond Column of _

this Table. '
That done, and the Equator drawn asbefore, if you would draw the

- Tropicks in che Polar Plane, look into the Table, and take 70 cent. out

of the Line of Inches, and prick chem down in the Subftylar, on either
fide of the Equator, and fo 72ceat. on the firft Hour, and 80 on the

:
‘

fecond :

ho ot gi

i and Circles of Declination: —— 63

’ + ‘
| fecond Hour, and 2 Inches H Ang. Po.) Tang. |_Secan. Trop;
gocent. cothe fifth Hour eee GE, M.In. Pa.dn. Pa.[in. Pa;

| from the Subftylar, and {12 f.0 00 Or 61f0 70

the reft of thefe Diftances IL fi Wy eke) 4.3\L.* 63/0 72

ontheirfeveral Hour-liness |r0o 2/30 00 931. 85/0 80

| and then drawacrooked | 9 3145 OF 6112 2710 99

Line through all thefe | 8 4{50 02 79) 2211 40

Points, fo as it makes no 7 37s 06” 00’ 2112 70
MeAneles, theLincfodrawn 7 PRT CUS
hall be the Tropick required. | In likemanner you'may draw any other
- Parallel of Declination. NL RNIN

: /) CHA PNR

| To defcribe the Tropicks, and other Circles of Declination, in fach a Plane
| as is neither Equinoétial wor Polar. ;

Re Planes neither Equinoétial nor Polar, the Equator will bea Right
| Line, the Tropicksand other Parallels of Declination will be Conical
SeQions, fome of thém parabolical, fome eliptical, but \moft of them
_ hyperbolical. ee ET ier Berd Oe, 5,
_ Tofind the Points of Interfection of thefe Parallels with the Hour-
| ines, we are to confider,
i
Inches, ial : '
| Secondly, The height of the Style above the Plane. ,
Thirdly; The Angles ac the Pole between the proper Meridian and the
- Hour-circles. sage | ak
Thefe being known, will help us cofind, firft, the Angle between the
Axisand the Hour-lines on the Planes amd then the diftance between the
~ Eenter and the Parallels. Both thefe may be reprefented in this manner,
Let the Triangle A B C be made equal to the Style belonging to your
Plane, A C the Subftylar, BC the Axis of the Style, A B the length of

the Style perpendicular to the Plane. Then having drawn the LineB D-

| perpendicular to the Axis on the Center B, and any Semidiameter B D,
- defcribe an occult Ark of a Circle, and therein infcribe a Chord of 23 gr.
30m. from D unto T, on either fide of the Line, with fuch other inter-
mediate Declinations as youintend to defcribe on the Plane ; forthe Per-
| | pendiculas:

Pai

Firft, The length of the Axis of the Style in Inches and parts of

54 «TE Deleription of the Troploke

pendicular BD fhall be che Equator, and BT the Tropicks, and the _
other intermediate Lines the Parallels of Declination. Wherefore you”
may take out the diftance C V from the Center to the Equator, and prick
it down on the Subftylar of your Plane from the Center at C unto Visa |
fo the Line drawn through ¥,. perpendicular to your Subftylar, fhall be |
the Equator of your Plane. Va
That done, take the diftance of each Hour line between the Center
and the Equator of your Plane, and.prick them down ‘in the Equator of
chis Figure, from che Center at C, noting the place where they crofsthe
Equator,with the Number belonging to che Hour, and drawing the Hours |
lines from C, through the Lines of Declination, ey |
Or, having the Seéfor, you may draw an occult Line C E, perpendi-
cular to the AxisB C, and therein prick down the Tangentof theheight
of the Style above,che Plane, from CuncoE: Thendraw theLineEF
parallel co the Axis, croffing the Subftylar produced in che point F; this’
Line |)

q

Ae
- +

and Circles of Declination. — PT RBs

Line EEF will be the Line of Sines upon the Sector, and therein you may

prick down the Sines of the Complement of the Angles’act the Pole from
F toward’ F,--and draw-the Hour-lines-by-thofe Points through the Lincs
of Declinatioiis-fo the Angles at C, between the Axis BC and thofe

{ . Hour-lines, thall be the Angles between the Axis of your Style and the

Hour-lines in your Plane; and the feveral Diftances between the PointC
and the Lines of Declination, fhall give you the like Diftances between
the Center and the Parallels of Declination upon the Hour-lines in your
Plane. Upon this ground it followeth, |

1,. To proportion the § tyle unto the Plane.

Confider the height of the Style above the Plane, and the length of
the Subftylar between the Center and the Place which you intend for the
Tropick, If it-be the Tropick which is farcheft from the Center, add
113 gr. 30m. if the nearer Tropick, add 66 gr. 30m, unto the height of

the Style, the Remainder unto 180 gr. thall give youthe Alcicude of the
| Sun abovethe Plane, when he cometh to that Tropick. As in our Lati«
~ tude, the height of the Style above an Horizontal Plane is ¢ 1 gr. 30 me.
' add unto this 133 gr. 30m, the fum is 165 gr. which being taken out a)

180 gr. the remainder will be 15 gr. and fuch is the Altitude of che Sun

_ abovethis Plane when he cometh to be in the Winter-Tropick : But if -
_ you add 66 gr. 30 m. unto 51 &r. 30m. the remainder to 180 gr. will .

be 62 gr. Aynd fuch is the Alticude of the Sun in che Sammer-Tropick.

Then, |

As the Sine of 66 gr. 30m.
~ to the Sine of the Suns Altitude :
So the Length of the Subftylar Line,
tothe Length of the Axis of the Style.

Asin the firt Examples of the Declining Vettical, where. the height

-of the Style was found to be 34 gr. 33 m. and ishere reprefented before,

pag. 31. by the Angle BC ¢ ; add to this height 113 gr. 30 m, for the

AngleCB @, the fum will be 148 gr. 3 m. and the remainder to 180 Ore

will be 31 gr. 57 m.and fuch isthe Angle B SC of the Altitude of the

Sun above the Plane, when he cometh to be in the Tropick of $,. which is
here the farcheft Tropick from the Center.

+ Then fuppofing the length of the Subftylar-line between the Center
hat lii

ik and

‘x
8

x i ae ee

and the Place which is fic for the fartheft Tropick, to be abou: 21 Inches,

extend the Compafies from the Sine of 66 gr. 30 m. unto the Sine of 3 rgr.
57 m. the fame extent willreach in the Line of Numbers from 21 unto

42.11, and fo the length of the Axis of the Style fhould be s2énch.11 cente — |

Or it may fuffice to make it juft 12 Inches,as. a more eafie ground for the
reft of the Work. |

But if ic were required to proportion the Style unto the Plane, fo as it;
may caft the Shadow to the full length of che Subftylar-line at all cimes of

the Year, you may then confider the Sun in the Tropick,which isto be fee
nearcht unto the Center, and add 66 gre 30m, unto 34 gr. 33 m. fothe

remainder

gd Civeles of Declination. 67

remainder unto 180 gr. will be 78.gr. 57m. And if you extend the

Compaffes from the Sine of 66 gr. 30.7. unto the Sine of 78 gr. 57 ms.

the fame extent will reach in che Lineof Numbers froma 21 Unto 22, 47

| for the length of the Axisof the Style.
; 3. Having the length of the Axis, and the height of the Style above the

Plane, to find the length of the Sides of the Style.

The Style of a Plane neither Equinoétial nor Polar, may be either a

{mall Rod of Iron {ec parallel to the Axis of the World, or perpendicu-
lar to the Plane, or elfe a thin Plate of Iron or Brafs, made in form of a
ReGangle Triangle B A C, with the Bafe B C parallel co the Axisof the
World, theSide AB perpendicular to the Plane, and the Side A C che
fame with the Subftylar-line; wherein knowing BC, and the Angle

I BAC,

oAs the Sine of 90 gt.
tothe Length of the eAxis:
Sothe Sine of the Height of the Style,
to the Length of the Perpendicular Side :
And {othe Co fine of the Height of the Style,
tothe Length of the Subjtylar fide.

~ Thus in the former Example, the length of the Axis being fuppofed to _
be 12 Inches, and theheight of the Style 34 gr. 33. Extend the Com

paffes from the Sine of gogr. (or elfe from the Sine of 5 gr. 45 #.) unto

12 in the Line of Numbers, the fame extent will reach from the Sine of

34 gr. 33m unto 6, 8o- in the-Line of Numbers, for the length of the
perpendicular Side ;. and fromthe Sine of g5gr. 27 m.unto 9.88 for the

length of rhe Subftylar fide.

3. To find the Diftance between the Center and the Equator upon the Sub-
ftjlar Line. |

This is here reprefented by C V7, and may be found by refolving the |

Reétangle Triangle CB.

Asthe Sine of the Height of the Styles
is tothe Sine of 99 gr.
Sothe Length of the Axis,
Tothe Diftance of the steel froms the Center
11 2

Extend

ae

68 _ The Defcription of the T: vopicks

Extend the Compaffes from the Sine of 55 gr. 27 m. unto the Sine of
_ .gOgr. the fame extent willreach inthe Line of Numbers from r2unto

X4.17. Wherefore if you take 14 inch, 57 cent. and pricking them

down on your Subftylar- line from C unto y, draw a Linethrough 7, — |

croffing the Subftylar at Right Angles, the Line fo drawn {hall be the
Equator. - *

4 To find the Angles contained between the Equator and the Hour-lines f

apon your Plane.

Thefe Angles made by ¥ and the Hour-lines are Complements of thofe
which are at C, between BC the Axis and thofe feveral Hour-lines, and

depend upon the Angles ar the Pole, between the proper Meridianand the _

Hour-circles,

“4s the Sine of 90 er. ;
to Co-fine the eAngle at the Poles
So the Co-tangent of the Height of the Style,

tothe Tangent of the Angle between the Equator and the Hour-lined |

Tn our Example the height of the Style is 34 gr. 33 m, and the proper
Meridian falleth to be the fame with the Circle of the fecond Hour after

Noon; whereupon the Angle at the Pole, between this proper Meridian, |

and the Circles of the Hour of 1 onthe one fide, and 3 on the other fide,

will bers gr. So between this Meridian and the Hour-circles of 10 and ~ |

4, the Angle will be 30 gr. ec. as in the Table,

An. Po [Ar.PiaeAn.to {> oC B/C
: Gr M (Gr, M,Gr. Mdin oP. In, Pin P.!
(Subfty].| © Cc] O O55 29114 57-20 8oj a 21
tT 3yts of 8 3854 3c/'4 F42n 36lrn 25
Hig aratzO0! OFS Se Roe 3°15 33,23 galti go
rcbe 2143. 1; S129 33-45. 45 bO" 75 29 Ob t7e
10 6/60 0/44 3036 = «6 o}'0. C50 Ba} t2 77,
9 FITS 64 4220 36/34 olInfinic. [rg 82
18 8190 190 QF Oo — cf{nfinir, 27. 60

a

Anof- |

: Tf then it be required to find the Angle which the Hour-line of 4 after f |
~ 9oon doth make with the Plane of the Equator, that is the Angle C 4.B,

contained.

-

é

and Circles of Declination. == 69

Sleitained- between the Hour-lineG 4 and the Line B.4, drawn from the.

op of the Style unto the Interfeétion of the Hour-line of 4 with the
equator, ; ’
‘Extend the Compatffes from the Sine of 90 gr. unto the Sine of 60 gr.
he Complement of the Angle ac the Pole, the (ame extent will reach from.

into the Tangent of 51 gr. 30m, and {uch is the Angle C 4B in the
Diagram,

_ Orin Crofs-work, if it were required to find the Angle C 9 B, look
nto the Table for the Hour of 9, and there you fhall find the Angle at
he Poleto be 75 gr. and if you extend the Compaffes from the Sine of
7O gr. unto the Tangent of 55 gr. 27 m. the fame extent will reach from

;|

36 m. and fuchisthe Angle C9 B, made at the Equator between the
Line B 9, drawn from the top of the Style, and the Hour-line C g, drawn
rom the Center. The like reafon holdeth for the reft, which may be

x elfe proceed,

5. To find the Diftance between the Center and the Parallels of De-

I: clination.

| The Diftances berween the Center and the Parallels of Declination:
~nay be found, by refolving the Triangles made by the Axis B C, the Lines
of Declination, and the Hour-lines. For having the Angles at the Equa-
or, and knowing the Declination of the Parallel, if the Parallel hall
all between the Equator and che Center, add the Declination unto the
Angle at the Equator: or if it fhall fall without the Equator, take the

Declination onc of the Angle atthe Equator, fo thall you have the Angle.

atthe Parallel. Then,

4s the Sine of the Angle at the Parallel,.

| to the Co-fine of the Declination:

— Sothe length of the Axis of the Style,

tothe Diftance Letweea the Center and the Parallel,.

| Thusin our Example,the Angle at the Equator belonging to the Foe

he Tangent of 55 gr. 27m. the Complement of the height of the Pole,

he Sine of 15 gr. the Complement of 7§ gr. unto the Tangent of 20 gre ,

ound and fet downin a Tables Then may you either draw thefe Angles.
it Cin the former Figure more perfe€tly, and thence finith your Work, _

of 4, after-noon was found before to be §1 gr. 30m. if you would find’
; the

~

“go ———«~«iTifevipnio of the Tropicks =

unto the Sine of 66 gr. 30m. the Complement of the Declination, the |
{ame extent will reach in the Line of Numbers from 12 unto rr, 40, and

take the Declination cutof the Angle atthe Equator, fo the Angle at the

holdeth for all the reft, which may be gathered and fet down ina Table, |

‘the Subftylar as before: And the like reafon holdeth for pricking down

the diftance between the Center and the Equator, extend the Compaffes |
from the Sine of 5tgr. 30a. unto the Sine of 90. gr. the Complement
of the Declination, the fame extent will reach in the Line of Numbers |
from 12.unto 15. 33, and fuch is the diftance u pon the Hour-line: of 4 |
berween che Cencer and the Equator. . !

fF you would find the diftance upon this Hour-line between the Center \/
and: the inner Tropick, whofe Declination is-known to be 23.97. 30m, |
add the Declinatien tothe Anele at the Equator, fo the Angle at the Pas ’
rallel will be 75 gr wherefore extend the Compafies from the Sine of 7 5.¢r.

fuch is che length of the Hour-line of 4 between the Cencer and the Tro-)
pick of vp. |

If you would find the diftance upon this Hour-line between this Cen
ter andthe Tropick of , which 1s here che fartheft from the Center, ,

Parallel willbe 38 gr. wherefore extend the Compaffles from the Sine of |
28 unto the Sine of 66 gr. 30m. the fame extent will reach in the Line
of Numbers from 12 unto 23.44, and fuch is the diftance between che
Center and Tropick of upon this Hour-line of 4. The like reafon’

That done, and the Equator drawn as before, if you would draw the |
Tropick of %, look into the Table, and there finding under the Title |
€ s the diftance of the Subftylar between the Center and the Parallel of |
3 to be 20 inch. 80 cent. take 20 inch. 80 cent, out of the Line of Inches, |
and prick them down in the Subftylar of your Plane from C unto g

Or if either the Center fall without your Plane, or the extent be too
large for your Compaffes, you may prick down the difference between |
C yandC g: As here the diftance C y~ between the Center and’ the |
Equator is14. 57, the diftance C & 20. 80, the difference 6.23, There=
fore taking Y 6 inch. 23 cent. prick them down onthe Subftylar from 6
unto %, and you fhall have the fame Interfection of the Tropick and)
of the re{tof chefe Diftances on their feveral Hour-lines, I

Then having the Points of Interfection between the Hour-lines and the —
Parallel, you may joyn themall in a crooked Line, without making of
any Angles, the Line fo drawn fhall be the Tropick required. And after
this manner you may draw any other Parallel of Declination, whereol
you have Examplesin moft of che former Diagrams. "i

, ie
f

“ ‘ ~

4 mt iz i 0. ¢.. 7 “Fe

| Oo and Cireles of Dechpations 7%

|

li 3 | .

ie ; | CHAP. XIV.

To defcribe the Parallels of the Sines in any of the former Planes,

| ; r
“Pye Equator and the Tropicks before defcribed do thew the Suns ea- | ,

| trance into 4 of the Signs; the Equator into Y and x», the one
Tropick into @, and the other into w: The reft of the intermediate
Signs will be defcribed in the fame manner asthe Tropicks,if firft we know
icheir Declination. :

| The manner of finding the Declination , not oncly of the beginning |
hd the Signs, but all other Points of the Beliptick, is before fet down in one
|2 Prop. Affronomical, by which you may find the Declination of thebe- |, 9 as
‘ginning of &, M,andm, to be 11 gr. 30 m. and of IL, &. #and as. tai

i

of 20 gr. 12m, intot

to be 20 gro 12m. If thenyou infcribe the Chords of 11 gr.30 m. and

he former Figure B DT, pag.64.from D toward

|the Signs required.

of the Axis, you may find the Angles at the

ip the Lines drawn from Bthrough the Terms of thefe Chords thall be
|

~ And with thefe Declinations, the height of the Style, and the length
Parallel, and then the Di-.

which being pricked down |

"4

“ftances beeween the Center and the Parallel,
upon the feveral Hour-lines, thall give you the Pointsof Inrerfeftion, by.
which you may draw the Parallelsof che Signs, asin the Figures belong-
ing tothe Polar Planes:

CHAP. XV. mo ae
‘To defcribe the Parallels of the length of the Day in-any of the

former Planes.

MPAHe lenecly of the Day will-always be 12 Hours, when the Sun 4
| cometh to be in the Equator, and this holdech in all Latitudes ::
Ducat other cimes of the Year the fame place of che Sun will not give the
fame length of the Day in another Latitude 5, wherefore the Latitude be--

ing known, we are firfty

| To o c 4

7 kay ; Bese ; in act : " eee Pasa § s Li a
ge Parallels of the length of the Day.
To find the Declination of the Sun agreeing to the length af the Dy. Be |

|

|

As the Sine of 9° gr. axes ; ‘|
isto the Sine of half the difference : ae
Sothe Co-tangent of the Latitude, ee me li
tothe Tangent of the Declination. = 7 |

As if the length of the Day propofed were tg Hours, the different
between this and an Equinoctial Day (whofe length is always 12 Hours):

A fex
if

bet : would be three Hours(which make 45 gre )and the half difference is 22 gr
30 wm. Wherefore extend the Compafies from the Sine of 90 gr. unto the —

Tangent of 38 gr. 30m. the Complement of the Latitude, the fame ex- \

tent

ae

tent willreach from the Sine of 2 2gr.30m. untothe Tangent of 16 gr,

55. for the Declination of the Sun at fuch time as the length of the -

Day is either 9 or 15 Hours; and from the Sine of 30 gr. unto the Tan-
gent of 21 gr. 40 m. for the Declination belonging to 8 or 16 Hours; and
from the Sine of 15 gr. unto the Tangent of 11 gr. 38 m. for the Decli-
nation belonging to 10 or 14 Hours; and from. the Sine of 7 gr. 30 ms.
unto the Tangent of 5 gr. 56 m, for the Declination of the Sun when the
leng:h of the Day isexcher 12 or 13 Hours.

~ Tf then you infcribe the Chords of thefe Arks into the former Figure
3DT, the Lines drawn from B through the Terms of thefe Arks fhall

kKkk be ©

Paraliels of the length of theDay. | * FB.

se - Unequal Planetary Hours. ©

which you may draw the Parallels of the length of the Day, whereof

Plane in Chap. 4.....And by, the fame reafon you may draw the Parallels |

> - (and by fomethe Planetary) Hours.

A af = ¥ 4 “ « wath

be the Lines belonging to the Diurnal Arks, and-the feveral Diftances |
becween them and the Point C, give the like Diftances berween the Cen-
cer and the Parallels of the lengch of the Day upon the Hour-lines in |
your Planes ** #75 . Tot a Op Ne lee

Or comparing thefe\ Angles of Declination with the Angles atthe
Equator, you may have the Angles at the Parallel, and then find the
Diftances between the Center and the Parallel, which being pricked down.
upon the feveral Hour-lines, fhall give you the Points of InterfeCtion, by |

you have anothersExample in the*Diagram belonging-toan- Horizontal.

of thofe Circles to which the Sun is Vertical, che Parallels of the princi
pal Feaft, or what elfedepends on the Declination of theSun,.

4 }
- -

. wu 2%

CHAP. XVI.
To draw the Old Unequal Hours in the former Planes.

ig was the manner of the Ancients to divide the Day into 12 equa

Day and Night into 24 Hours. Of thefe 24, thofe which belonged’
untothe Day were either longer or thorter (excepting the two Equi a
étial Days) than thofe which belonged untc the Night; and’the Sum’
mer Hours always longer than che Hours inthe Winter, according to th’
Jengthning of che Days, whereupon they are called the Old Unequa

To exprefs thefe in the former Planes, firft draw the common How
Innes, the Equator,/and the Tropicks, as before: Then ee
cult Parallels of the length of the Day, one.for @ Hours, the other fe
15 Hours; for fo you may draw a flraight Line for the firft uneau:

Hour through § 40. 45 m. in the Parallel of 15, and through 8 ho. i
inthe Parallel of 9, This ftra'ght Line’(hall pals direQly through 7h
@ #. in the Equator, and fo cucoff a twelfth pare of the Arks above cl
Horizon, both frem thefe two Parallels and the Equators and beir

continu!

cad

’ oF
* sd
Me ON OD
s i .

aS

Mo cut off about a twelfth part from
£D. and all che reft of the Parallels
)

‘rrour.

‘cond unequal Hour through 7. hoxr

n the Equator, and through 9 bow
nthe Parallel of 9, .and foin the reff,

sin this Table,
_ And of thefe :

1ave a further Example im the Dia-

| ATE re ’
ne
| oP

2 4
-e'tgseoursb som

unto the Tropicks, ic thal”

f Declination , without any fenfible.” |
In like manner you may draw the: .
nthe Parallel of 15, through 8 bivr |
inequal Hours you “|

yam belonging co the Polar Declining , |

‘te -

Pa
Gas en eee

~

[erendcu sen ranacusseyceaneaay Gearewe qrarte
: :
. a

gk «sets fois Sun-vifag and Sam-fetting) =

CHAP. XVII. iw mmid
To draw the Hours from Sun-rifing and Sun-fetting in the fi ormer Plants. q

4) fe know how.many Hoursare paft fince the Sun-rifing, or how ma- |
. ny remain tothe Sun-fetcing, firft draw the common Hour-lines, «
the Equator, and the ,Fropicks, as before; chen defcribe cwe occule Pa-

<= 8 A | 7m “i Ay a pe

J

c¢

-rallels of the length of the Day, one for $'Hours, and the other for 16
Hours: For foyou may draw the firft Hour from the Sun-rifing rhrough —
the common Hoursof 5 inthe Parallel of 16, of 7in the Equator, and

of |

"To draw the Bovizintal- line: 97
_ of gin the Parallel of 8. ‘In like manner, the fecond Hour from Sun-

rifing, through the common Hours of 6in the Parallel of 16, of 8inche
Equator, and of ro in the Parallel cf 8. And fothe reft intheir orders
- The firft Hour before Sun-fetting, or che 23 Hour from the laft-Sun-

~ fecting, may be drawn in like fort, through the common Hours of 3 after-

noon.in the Parallel of 8, of § inthe Equator, and of 7 in the Parallel-of
16. The fecond Hour before Sun-fetting, or the 22 Hour after the laft
Sun-fetting, through the common Hours of 2in the Parallel of 8, of 4
in the Equator, and of 6 in the Parallel of 16: And fo thereft in the

like order, whereof you have another Example in the Diagram belonging

tothe Declining Vertical,

CHAP. XVHK

To draw.the Horixontal-line in the former Planes:

i a Bee common Hour-lines do. commenly depend on the fhadow of the

Axis; but che Parallels of the Signs, and of the lengch of the Day,
the Hour-lines from Sun-rifing and Sun-ferting, with many others, de-
pend on the Shadow of the cop of the Style, or fome other Point in the
Axis, which here fignificth the Center of the World, and is -reprefented
by che Point B. And thefe Lines fo depending are thefi Onely ufeful,
when they fall becween the two Tropicks, and within che Horizon.

There may be feveral Horizontal-lines drawn upon every Plane, as I
fhewed before in finding the Inclination of a Planes but the proper Hori-
zontal-line,which is here meant, mutt always be inthe fame Plane with B
the copof the Style; fo that inan Horizontal Plane chere-can be no fuch
Horizontal-line: but in all other Planes it may be found by applying the

_ Horizontal Leg of the Se@tor unto the top of the Style, and then working
_as before; and.the Interfection of this Line with the Meridian or Subfty-
lar-line may be found by Proportion.

1. To find the Inter/eGtion of the Horizon with the Meridian in an Egut-
notisal Plane.

eAs the Tangent of 453-2.
to the T.angent of the Latitude ¢
So is the Height of the Style,
to the Diftance between the Style and the Horlzontal-Line.
i As

ey pe

a VE 2 Gah he). | RON Sra a sc re es
. ae es BAe. a ; ;

- Asin the Example of the former Equinoctial Plane, extend the Com- |
pafles from the Tangent of 45 gre unto 51 gr. 30m, the Tangent of the.
Latitude, the fame extent will reach in the Line of Numbers from-sr .
“ the length of the Stylé, unto 66, and fuch is the Diftance between the
~ Style and the Horizoncal-linc: Wherefore I take 66 parts ouc of a Line
of Inches, and’ prick them down in. the Meridian-line from C unto H
above the Style in the upper Face, but below the Style in the lower Face

78 9 draw the Horizontal-lings = ,

of the Plane; foa Right Line drawn through H, parallelto the Hour of
6, thall be the Horizontal-line. a | 2

2. To find the Inter{ettion of the Horicon with she Aderidian in aDireét
‘Polar Plane. .

As she Tangent of 45 gt
to the Co-tangent of the Latitude :
So the length of the Style, | |
to the diftance betweenthe Style and the Horizontal-line.
As in the Example of the former Polar Plane, extend the Compafles .
from the Tangent of 45:.gr- unto the Tangent of 38 gr. 30 m.cheComs
plemenc of the Latitude, the fame excent will reach inthe Line of Num= *
~ bers from x. 61 the length of-the Style, unto.1.28, andfuchis the di-
{tance upon the Meridian becween the Style and the Horizontal-line. iy
In all upright Planes, whether they be Dire&, Vertical or Declining,
or Meridian Plane:, the Horizonral-line muft always be drawn through
_ Athe Foot of the Style,, as may appear in the Examples before. ee
And generally, in all Planes whatfoever, the Horizontal-line muftbe
drawn through the Interfection of the Equator with the Howrof 6. Or
if that InterfeCtion fall wichout the Plane, yetif any Arksof thelengch —
of the Day be drawn on the Plane, the Horizontal-line may be drawn
chrough cheir Incerfections with the Hours of che Suns rifing or fetting.

a ett Wr > = _ oe —

Kae

nere. mt ' :
~ <Confider the length of the Style in Inches
the diftance of each Azimuth from the Style, according to the Angle at

ia ae i >

—- The Deleription of the Vertical Circles. 79

CHAP, XIX,

To deferibe the Vertical Circles in the fortser Planese

>

He Vertical Circles, commonly called Azimnths, are Great Cir-
cles drawn through the Zenith, by which we may know in what
part of the Heaven the Sun is, how far from the Eaft or Weft, and how

near unto the Meridian. |
~ In allupright Planes, whether they be Dire&t Verticals, or Declining,

| or Meridian Planes, the Semidiameter of che Horizon will be the fame
_- with A B the perpendicular fide of the Style, and chefe Azimuths will

be Parallels.one to the other, and the diftance of each Azimuth from the
Foot of the Style upon the Horizontal-line, may be found in this man-

and parts of Inches, and
the Zenith in Degrees and Minutes.
As the Tangent of 45 gr.

- ¢0 the Tangent of Azimuth:
So the length of the Style,

to the length of the Horizontal-line between the Style dud the

Aximath.

As if it were reditired to draw the common Azimuths on the South
Face of the Vertical Plane before defcribed, where A B the length of the

Style may be fuppofed tobe 10 Inches, |
Here the Plane having no‘declination, the Style is in the Plane

of the Meridian, ‘and fo poincech direétly into the South, The

Point of SHE is 11 gre 15 wa diftant from che Style, and SS £
22 gr. 30m, and the reft in their order: Wherefore extend «the
Compafies from the Tangent of 45 gr. unto ro in the Line of Num-

bers, the fame extent will reach from the Tangent of It gr. 15 ™
: unto

eee =

~ The Deferiprion of the Azimuthsy

MS 7

sto
ACs

| \

er

\

-
*

j- jAn.Zen,| Tang. | nto %,99 in the Line of ‘Numbers for

Gr. M.j[n,_ Pa.| the length of the Tangent-line, between
: }o of © of the Style and the Pome S6E; and
II 15/1 99) from the Tangent of 22 gr. 30 m. unto

22 30| 4 %I4) 4.14 for SSE: And fo for the ret, }
45| 6 68] asin this Table. \

143 O10 ©0 In like «manner, ‘in the firft Exam-
| 15/14 971 ple of the Declining Plane, where the
30.24 14] Style ftandeth according to the Declina-

78 45|50 27] tion 24 gr,20 m. diftant from the South
Eaff, 99 Olnfinic.| toward the Weft, the next Point of

SOW

|

|

(gom. and the third of SWbS is again

j- fAn, Zen. Tang.
.|Gr, M.In. Pa.

9 gr. 25 m. and the reft in their order. [524/89 394 00
Wherefore having before found the length SE {69 2018 03
“of the Style to be 6 Inches 80 parts, ex. [S858 = 10 on
tend the Compaffes from the Tangent of |> 5% |46 50) 7 25
45 gr. unto 6.80 parts in the Line of S6E 135 3314 86
“Numbers, che fame excent will reach from |$0#! [24 20) 3. o7}
the Tangent of 24 gr. 20 m.unto 3.07 in Feeble 5) 1. 5 8

.

but fo. as they may be contained beeween /

the Line of Numbers, for the length of f{2 SOO f 22
the Tangent-line beeween the Style and the The| Foot of \the Style
South; asd from che Tangent of 13 gr. J°#S| 9 25) 1 13h
ly m. unto ¥. §8 for che Point of ShW: |S¥...|20 40] 2 §7
and fo-for the reft; asin this Table. SWOV}31 55] 4. 24
That done, if you take thefe Parts our [VOW |43. Ic} 6 375-
of a ‘Lineof Inches,and prick them down = |” bS |54 25) 9 $0
inthe-Horizoncal-line on eitherifide of the © (Wee 65 dots 02
Style, drawing Right Lines perpendicular pe 72 _ SS|T9 26
‘o the Horizon through thefe InterfeCtions, WEN RIES rola 45

he Horizontal and the Tropicks, the Lines fo drawn thal! Lethe Azi-

“nuths required. | ;

Inian- Horizontal Plane thefe Azimuths are drawh more eafily: For

ere the perpendicular fide of the Styleis the fame with the Axis of the
forizon, and the-Foot of the Style is theVertical Point, in which ak

he Azimuth-lines dommeer, as their'Circles do in the Zenith: Where=

ore let any Circle defcribed on the Center Ay at the Foot of the Style, be
livided firft into four parts, beginning at the-Meridian 5 and then cach
juarter fubdivided’either into eight equal parts, according to the Points.

if the Marinets Compafs, or into 90 gr. according to the Aftronemical
livifion’s if youdraw Rjght Lines through the Center and thefe divifi-
ms, the Linesfodrawn fhallbethe Azimuthsrequired,

| Inall other Planes inclining to the Horizon, thefe Vertical Circles will
neetina Point ; but chat Vertical Point being more or lefs diftant from
he Foot of the Style; the Anglesat this Point will be uncqual,

) 5 e435 4 yi

Rl aan sy ge

; upon a0 Inclining Plage, Sy
‘SbwWisbut 13 gr. 5 m. diftant from the
‘Style; and the fecond of SS wonely I gr.

a: The Deferiptien of the Azimuths —

x. To find the diftance ietmirs the oat of the Siyle 4 the V erlieal S|
Point. |

The Vertical Point, wherein all the Vertical Lines do meet, “will be ale
ways in the Meridian,direétly under or over the top of the Style; and the
Angle between the perpendicular fide of the Style, and the Vertical line,
will be equal co the Inclination of the Plane to the Horizon. Wherefore,

MIPS

bis As the Tangent of 45 er.
to the Tangent of i: Inclination of the Plane:
Sois the length of the Style, a
go the diftance betweenthe Foot of the Style and the Vertical Points |
Sah Vag lig | Tie

SS asi) Bee ~ vr Ss. Oe
aes ve ; : ‘ L 1 pe -
‘
|
‘

ca apo an Taclining Planes 83
_ Thus in the firft Example of the Declining Inclining Plane, where che

“upper Face of the Plane looking South-weft, che Declination was 24 gr.

20 m. the Inclination 36 gr. and you may fuppofe A B the length of the
Style to be 6 Inches; if you extend the Compafies from the Tangent of
45 gr. unto the Tangent of 36 gr. the {ame extent will reach inthe Line
_of Numbers from 6,00 unto 4. 36, for the diftance AV, between A
the Foot of the Style and V the Vertical Point.

=

“© 2, To find the diftance between the Foot of the Style and the Horizontat-
| line. ; |
.. Asthe Tangent of the Inclination of the Planes

is tothe Tangent of 45 gr.
Sothe length of the Style, | |
; to the diffance between the Foot of the Style and the Horizental-
ip ome line.

© So the fame extent of the Compaffes as before will reach in the Lire of
“Numbers from 6,00 unto 8. 26 for the diftance A H between the Foot
of the Style and the Horizontal-line.

"Then may you take 4 inch. 36 cent. and pricking them down from’ Av
| the Foor of the Style, unto V the Vertical Point in the Meridian, draw
the Line V A, which being produced, fhall cut the Horizon inthe Point
H withRight Angles, and be chat particular Azimuth which ts perpendi-
-cular cothe Plane. |

Or, you may take8 inch. x6 cent. and prick them down1in the former
“Line V A, produced from A unto H, and fo draw the Horizomtal-line
| through H, perpendicular unto V H, which Horizontal-line being pro-
duced, will crofsthe Equator in the fame Point wherein the Equator crof=
fetch the Hour-line of 6, unlefs there be fome former error.

3. To find the Angles made bythe Azinoath-lines at the Vertical Point.

ws =

The Angles at the Zenith depend on the Declination of the Plane, as
in our Example, where the Style*ftandeth according to the Declination
24 gr. 20 m. diftanc from the South toward the Weft, the Azimuch of
10 gr, from the Meridian Eaftward willbe 34.gr. 20 m, the Azimuth of
| 10 gr. Weltward willbe onely 14 gr. 20 m. diftant from the Style; and
| fo sherett in their order.

Lil 2 Or

——_ FF NLC UN, 23 oes es eee, CL 4 Fe OR eee ike Bole Bays el ame ay a” Pos to on Pe iA yf: ie Pe,

84 «= The Defeription of the Azimuths

‘Or if you would rather defcribe the common Agimuths, the Point of

SWE willbe 35 gr. 35 m. the Pointof SbW 13 gr. 5m. diftant from, —
! dd a

the Scyle ; and fo che reft in their order. Then, .
| Abs the Sine of 90 ere | ie

to the Co-fine of the Inclination of the Plane:
So the Tangent of the Angle at the Zenith,

to the Tangent of the Angle at the Vertical Point, between the Linea

drawn through the Foot of the Style, and the Axinsuth required.

the Sine of 54 gr. the fame extent thall

in this Table.

fcribe an occult Circle, and therein infcribe

VH, and then draw Right Lines through

Azimuths required.
The like reafon holdeth for the drawing-of the Azimuths upon. all

other Inclining Planes, whereof you have another Example in the Dia- 4

gram belonging to the Meridian Incliner, as before.

Or, for further fatisfaction you may find. where each Azimuth-line.

fhall crofs the Equator.

As the Sine of g0 gr.
te the Sine of the Latitude : |
Se

ma
‘-.

Wherefore the Inclination of the Plane |
in our Example being 36 gr. extend the
Compafies from the Sine of 90. gr. unto

reach inthe Line of Tangents from 24 gr.
20 m. unto 20 gr. 5 m. for the Angle |
HV aat the Vertical Point, between the
Line VH, drawn through A the Foot of —
theScyle, and che South. Again, the fame _
extent will reach from the Tangent of 13.
£r.§ m. anto 10 gr. 38 m. for the Angle
belonging to S b¥3 and fo for the reft, as

Thefe Angles being known, if on the -
Center V, at the Vertical Point, you de-

the Chords.of thefe Angles from the Line |

the Vertical Point, and the Terms of thofe |

Chords, the Lines fo drawn fhall be che

es

upon an Inchining Plane. -
Sothe Tangent of the Azimuth from the Meridian,
tothe Tangent of the Equator from the Meridian.

«by

Extend che Compafies from the Sine of go gr. unto che Sine of our

| Laticude 51 gr. 30 m. the fame Excent will reach in che Line of Tangents
__from 10 gr. unto 7 gr. §0m. for the Interfeétion of the Equator with ~

the Azimuth of 1ogr. fromthe Meridian. Again, the fame extent will
reach from 20 gr. unto lt 5 gr. §4 mm. for the Azimuth of 20 gr. And fo

the reft, as in thefe Tables.

Azim Equat, : Equat.
Gr. M. | Gr. M Gr. M. | Gre M.
TOO TOW 7 go Sy ae §1
202 O PIS? 54 22, 30/18 58
30 0424 20 33.6 BS | Az FO) o
40 2/8 18 45 Oh JO
50. 0143 oO JO 15 | 49 30
60 O53 35 67 301 62°" 6
70 (0 | 6S / bs 45 175 44
80° 0177 18 90 0 | 99 Oo
90 o|90: 0

By which you may fee that the Azimuth go-gr, diftant from the Meri-

dian, which isthe Line of Eaft and Weft, will crofs the Equator at go gr.

from the Meridian, in the fame Point with the Horizontal-line and the —
Hour of 6: and that the Azimuth of 45 gr. will crofs the Equator at 38’ gr.

Like m. from the Meridian; that isy the Line of S$ E will crofs the Equator

atthe Hour of 9 and 28 m. inthe Morning, and the Line of § W: at 2 bo.

32m.inthe Afternoon: And {fo for the reft; whereby. you may examine
your former Work,

CHAP. XX.
To deferibe the Parallels of the Horizon in the former Planes.

*# He Parallels of the Horizon, commonly called Almicanters, or Pa-
rallelsof Altitude (whereby we may know the Altitude of the Sun

above the Horizon) have {uch refpeét unto the Horjzon, as the Parallelsof

Decli-

ie. hon, .. te - Ee EO Ne ER Ae a OE na Len ee eae a tee ae

“96 The Defeription of the Parellel of the Horlzon .
Declination unto the Equator, and fo maybe defcribed in like manner)~
In an Horizontal Plane chefe Parallels will be perfe& Circles y where-

’ fore knowing the length of the Style in Inches and parts, and the diftance
* of the Parallels from the Horizon in Degrees and Minutes, ;

As the Tangent of 45 gt. |
is to the length of the Style:
S6 the Co-tangent of the Parallel,
to the Semidiamseter of bis Circle.

“Thus in che Example of the Horizontal Plane, if ‘A Bcthelensthof the ~
Style thall be 5 Inches, and chat ic were required to find the Semidiame-
cer of the Parallel of 62.-¢r. extend che’ Compaffes from the Tangent of __
“45 gr. unto 5. 08 in the Line of Numbers, the fame extent will reach
from the Tangent of 28 gr. the Complement of the Parallel, unto 2.65:
And if you defcribe a Circle onthe Center A, to the Semidiamerer of
~2 inch. 65 cent, it fhall be che Parallel required. eo |
In all upright Planes, whether they be Dire& Verticals, or Declining,
or Meridian Planes, thefe Parallels will be Conical Se@ions, and may be
~ . drawn through’ their Points of Interfection with the Azimuth-lines, in
the fame manner as the ‘Parallels of Declination through their Points of
InterfeGtion with the Hour-lines. To this end, you may firft find che —
- diftance between che cop of the Style and the Azimuth, and then the di-
ftance between the Horizon and the Parallel, both which may be repree
_ fenced in this manner. |
On the Cencer B, and. any Semidiameter BH, defcribe an occule Ark
of a Circle, and therein infcribe che Chords of fuch Parallels of Altitude
_as you intend to draw on the Plane, (I havehere put them for rg, 30, 45,
and 60 gr.) then draw Right Lines through the Center and the Terms of
- thofe Chords, fo the Line B H fhall be che Horizon, and the reft the Lines
of Altitude, according to their diftance from the Horizon. |
~ Thar done, confider your Plane, (which here for example is theSouth
‘Face of our Vertical Plane ) wherein having drawn both the Horizontal ©
cand Vertical Lines, as I {hewed before, firft cake our A Behe length of ©
~ the Style, and prick chat down in this Horizontal’Line from, B unto A 3.
then take out all rhe diftances between B the cop of the Style and the fe-
veral Points wherein the Vertical Lines do crofs the Horizontal, transfer —
them into this Horizontal-line BH, fromthe Center B, and at che Terms
~of thefe diftances erect Lines perpendicular to. the Horizon, noting chem
. | with

lhe.) Ss erg ; ¢ ,
io ~ , * -~

- “upon an Horizontal Plane. ) 87

aa eer
See

Rm.

HW .
TB scans Ae
. :

— wich the Number or Letter of the Azimuth from whence they ih bs
ken ; fothefePerpendiculars thall reprefent chofe Azimuths, and the fe~ -
veral diftances between the Horizon and the Lines of Altitude thall give:

the like diftances between the Horizontal and the Parallels of Altitude «
upon the Azimuthsin your Plane. Upon this ground it followecth,

‘1, To find’ the diftance between the top of the Style, and the feveral Points -
whereinthe Azimuths do crefsthe Horizontal-line.

Having drawn theHorizontal and Azimuth Lines as before, look into -
the Table by which you drew them, and there you {hall havethe Angles.at..
the Zenith. Then, : ) | +

As the Co fine of the Angle at the Zenith, .
js.t0 the Sine of 90. gre

So the length of the Style,
to the Diftance required, -

7 Azimuths.

we ert aa hoe 6 La ll bs sul Oe os

$8 The Defeviptiow of the Parallels of the Elovizon

;

atk

OP
7

Te

at
HAS
HEAT

\

TF

muths.iGr. M.\In.

7 So ee

Ang.Ze./Tangent] Secant. |Par. 15. 30.

JES 178 45\50 a7lyr 26l13, F329”
-|Eaf. 90 o-Infinie. Infinit. (Infinit. |Infinic. |

AA RCPS. ]
r oe

| er
iv

8

. _ aon an Inclining Plante — |
As in our Example of the Vertical Plane, where AB the length

_ of the Style was {uppofed to be 10 Inches, extend the Compaffes from
the Sine of 7g gr. 45 m. (the Complement of 11 gr. 15 m. the Angle ac

the Zenith, belonging to §4 E and SbW) unto the Sine of 90 gr. the

- fameexcent willreach from 10. e@ the length of the Style, unto 10. 20,

| _ for the diftance between the top of the Style and the Interfegtton of the
- Azimuth S 4 E with che Horizoncal line, which diftance may be called

the Secant of the Azimuth, and may ferve forthe drawing of the Paral-

‘Aclof 45 gr. from the Horizon, The like reafon holdeth for the reft of

thee diftances here reprefented in the Line BH.

aur. ie

ae

2. To find the diftance betwten the Horizon and-the Parallels.

4s the Tangent of 45 er. |
to the Tangent of the Parallel:
So the Secant of the Azimuth,
to the Diftance required. —

As if it were required to draw the Parallel of 1 5 gr. from the Horizon,

upon this Vércical Plane; extend the Compafles from the Tangent of

45 gr. unto the Tangent of 15 gr. the fame extent will reachin che Line
of Numbers from 10. oo the Secant of the South Azimuth, unto 2. 68,
and therefore the diftance between the Horizon and the Parallel of 15 gre

is 2 inch. 68 cent. upon the South Azimuth, Again, the fame extent

will reach ffom 10. 20 cheSecant of SbE, unto 2. 73, for the like die”
ftance belonging to S6E and SbW: And fo for the reft, which may
de gathered and fer down in the Table.

That done, and the Horizon and Azimuths being drawn, prick down

10 Inches from the Horizonral- fine upon the South Azimuth,and 10 isch.

20 cent, on the Azimuths ef Sb E and S LW, and 10 inch. 82 cent. on
the Azimuths of § § Eand S § W, and 12 inch, 3 cent. onthe Azimuths
of SEL Sand SWbS, and fo the reft of thefe diftances on their feve-

yal Azimuths: then if you draw a crooked Line through thefe Points,

that may make no Angles, che Line fo drawn thall be the Parallel of 45 gr.

_ from the Horizon. ‘In like manner may youdraw the Parallel of 15 gr.

or any other Parallel of Altitude, upon any Vercical Plane.

If the Plane incline to the Horizon; after we have found the Vertical
Point, and drawn the Horizontal-line, we are farcher to find che length

“of the Axis of the Horizon, then the Angles betwixt this Axis and the

Mmm Azimuth

go To draw the Parallels of the Horizon
Azimuthelines, and fo che feveral diftances between the Parallels and the
Vertical Point, all which may be reprefented in this manner.

Onthe Center B, andany Semidiameter, defcribe an occult Quadrane

of a Circle, and therein infcribe the Chords of fuch Parallels of Altitude

as you intend to draw on the Plane, drawing Right Lines through the -
Center and the Terms of thefe Chords, fo the Line B H thall be che Hori-. .

zon, and his Perpendicular B V the Axis of the Horizon, andthe reft the -

Linesot Alticude, according to their diftance fromthe Horizon. |
_ That done, confider your Plane, which here, for exampleyis che firft of
ef our three Declining Inclining Planes; wherein having drawn both

the Horizontal and Vertical Lines, as I fhewed before, firftrake out the -

Axis of the Horizon, which is the Line becween B the top of the Style and
V the Vertical Point, and prick thacdown in this Figure fromB unto

V5 then cake out both the Line V H, and all che reftof the diftances bee

tween V the Vertical Point and the feveral Points wherein the Vercical .

Lines do crofs che Horizontal Line of this Figure from the Point V, noting

the place where they crofs the Horizontal-line, with the Number or Letter

of the Azimuth from whence they were taken, and drawing the Azi- |

muth-lines from V chrough the Lines of the Altitude.

Or having the Se&tor, you may draw an occult Line V E er Vera
Tarcothe Axis V B,. and here prick down the Tangent of the Comp

“

rig |
-%
4

ment

;

~

sb apietes: ait a atti "hee | :
pens ane Zaclining Plant. ot

ment of the Inclination of che Plane from V unto E: then draw the Line

EF parallel co the Axis, croffing the Line V H produced in the Point F 5
fo this Line E F will be as the Line of Sines upon the Seétor, and therein
you may prick down the Sines of the Complement of the Angles at the
_ Zenith from E towards F, and draw the Vertical-lines by thofe Points
) through the Lines of Altitude ; fo the Angles at V, between the Axis
‘VB and thofe Azimuth-lines, thall be the Angles between the Axis of
the Horizon and the Azimuth-lines on your Plane, and the feveral diftan-
ces between the Point V and the Lines of Altirude fhall give the like
- diftances berween the Vertical Point and the Parallels of Alcitude upon
the Azimuths in yourPlane. Upon thisground it followeth,

1. To find the length of the Axis of the Horizon.

The Vertical, Point is always either direlly over or under the top of
the Style, and the diftance between them is that which I call che Axis of

‘the Horizon, which may thus be found ny oa

As the Co-fine of the Inclination,
to the Sine of 90 gr.
-Sothe length of the Style, sith Pox
to the length of the Axis of the Horizons
For example, in the firft of the three Declining Inclining Planes, the
Jaclination to the Horizon is 36 gr. the length of the Style A B 6 Inches;
extend the Compaffes from the Sine of 54 gr. the Complement of the In-

dination, unto the Sine of 96 gr. the fame extent will reach in the Line
of Numbers from 6. oo unto 7.42 5 and fuch is V B the length of the

‘Axis required.

2, Te find the eAngles coatained between the Horizon and the Werticat
Lines upon onr Plane. bee ii

a
de

The Angles at the Vertical Point between the Axis of ae Horizon and

the Azimuth-lines upon your Plane, are reprefented in this Figure by.

thofeat V, between V.B.and the Azimuths,. The. Angles between the
Horizon and the Azimuth-lines being Complements to the former, are
reprefenred eicher by thofe which are made by V E, or by BH, and rhe
Azimuth. lines which are drawn from V. a.

Mmm 2 That

en ae

; é Dhan ot tae ORO aEeS Ree ee 1 eS ee =e
/ ar ; P aN we aay
s aN & , * : :
\ ¥

92 The Defcription of, the Parallels of the Horizon = —
That you may find them, look into the Table by which £6 drew the —
rt ‘ . ‘

Azimuth-lines, there fha!l-you find che Angles act cheteonich, Then,
“As the Sine of 90 gr. aiid
tothe Co-fine of the Angle at the Zenith:
Sothe Tangent of Inclination to.the Horizon,

to di z angent of the Angle between the Horizon and the Vet |
: i

“Tnourexample, where the Inclination to the Horizon is 36 gr. and the
_ Angleatthe Zenith, between the Azimuth ac che Style and the Meridian,

is according to the Declination 24 gr. 20 m. extend the Compafies Pa a
ehe Sine of 90g. nnto the Tangent of 30¢r, che fame extent will reach
¢ oe i "from.

We Ang.Ze.] Ang. V.{An. Ho.| Horiz {11 18/26 34145 Oo
| | muths. \Gr. M.|Gr. M.|Gr. M.|In, Pa.lIn. Pa.lfn. Pa.lins Pav
| Eft. |t14 20\119 12/16 40 Inf. 38. GolIl oF
(sees FOF 9 $1 1060249 220 nite 210 24|22 401 9 oOo
(ESE lot soloz 16) t 20] "Jar o8lry 5717 60
(| SEDLE |80 35178 as| 6 47/62 82/23 4ali2 o7| 6 68
SE 69 20/6§ O]14 23129 87116 79/10 %12! 6 oo
SEbS |\58 5|52 25121 0,20 70113 611 8 ool 5 oh
SSE |46 50/40 46/26 25/16 68{1r go] 8 37 5459
{SFE 135 35130 3/30 35/14 §8110 go] 7 gol 5 42
| et 24 20/20 §/33 30113 44)10 eh a 35]
([SbwW |13 §|t0 39/35 17]12 84l10 2] 7 SiS coon
[SSH [1 sof t 20/35 sojtz_ 62] 9 90} 7 45 Sey
| eae | Size. {O° 0130 ONS “621, 4 : 7 : $ a3
‘[SWes | 9 251 7. 38/35 37/12 741 9 90] 7 sol 5 32h
SW 20 40116 $8134 12]13 20]10 20] 7 sot 5° 34h7
[SWE 13t 55/26 45/31 46/14 13/10 67/7 811 3 30
|W SW |43. 10/37 11/27 §sits B5jr1 So] 8 15]~5° “49
TWhS [84 25/48 30122 ssl19. osl12 04] 8 73 5 66
|Weff 165 . 40/60 48116 goj25 87]19 gt] 9 6o 5 96
THEN 176 55173 $81 9. 20/45 75\20 Ggltr 3216 4oy
wNw\88 10 37 44. t 20/318 88/33 27/14 18) 7 25
{Nvbwigg 25\101 35] 6 47| Infi- 192. 40!19 60\ 8 48
~NW [10 40\115 oof14 23] nice. | 31 4/10. 30
| Then may you either draw chefe Angeles at V in the former Bioure.

_ more perfectly, and chence finith your Work, or elfe-proceed..
4 33 Te:

I "upon an Inclining Plane.
_ from the Sine of 65 gr

93

. 4.0 2. the Complement of che Angle at the Zeniti,
unto the Tangent of 33 gr. 30m, fer the Angle contained between the.
| Horizon and the South part of the Meridian-line. Again, the fame ex-
_tenc will reach from the Co-fine of 35 gr. 35 ms, the Angle at the Zenith
‘belonging to SLE, unto the Tangent of 30gr. 3m. for the Angle be-
tween the Horizon and the Azimuth-line of SBE. The like reafcm
holdeth forthe reft, which may be found and fet down in the Table.

of To deferibe the Parallels of the Horizon

3. To find the diftance Letween the Vertical Point, andthe Parallel of |

© the Horjxon. 2 Sipps

. & Thefe diftancesmay be found by refolving the Triangles in the laft Fis

~ gure made by the Axis, the Lines of Alcicude, and the Azimuth lines.

For having the length of the Axis, and the Angle at the Horizon, if you

add the diftance of the Parallel from the Horizon, unto the Angle at che
Horizon, you thall have the Angleat the Parallel, Then,

eA the Sine of the Angle at the Parallel,
to the Co-fine of the eAltitude :
So the length of the Axis,
to the difance between the Vertical Point and the Parallel.

Thus in our Example, if it were required to find the diftance upon the —
Stylar Azimuth V H, between the Vertical Point and the Horizon, you.

have the Rectangle Triangle V BH, wherein che Angle ac the Horizon

here reprefented by B H V is (equal to the Inclination of che Plane) 36 gre

and B V the Axis of the Horizon between the Plane and the top of che
Style is-7 inch. 42 cent. Wherefore extend the Compafies from the Sine

of 36 gr. untothe Sine of 90 gr. the Complement of the Altitude, the —
fame.extent will reach in the Line of Numbers from 7. 42 unto 12,623
and {uch is the diftance of the Perpendicular Azimuth-line V H, between |

the Vertical Point and the Horizon.

In like manner, 1f you would find the diftance upon the Meridian |

between the Vertical Pointe and the Horizon, extend the Compafies from
theSine of 33 gr. 30m. the Angle at the Horizon, tothe Sine of 90 gr.

the fame extenc will reach in the Line of Numbers from 7.42unto 13.44, _
and {uch is V 4 the diftance becween the Vertical Point and che Horizon —

upon the Line of the Souch Azimuth, that is, upon the Meridian-line,

eS EEE EE ee eee

But if you would find the diftance upon the Meridian between the

Vertical Point and any other Parallel of the Horizon, as upon the Paral-

lel of 26 gr. 34m. then add thefe 26 gr. 34 m. unto 33 gr. 30m. the

Angle at the Horizon, fo thall you have 60 gr.4m™. for B DV the An-
gle ac the Parallel. And if yon extend che Compafles from the Sine of 60 gr,
4m. unto the Sine of 63 gr. 26 m. the Complement of the Parallel from

the Horizon, the fame excent will reach in the Line of Numbers from |

7-42 the length of the Axis, unto 7, 66, and fuch isthe diftance V D

berween |

|

ups an Inclining Planes 9%

between the Vertical Point and the Parallel of 26 gr. 34 upon the Me-
“tidian-line, The like reafon holdeth for all che reft, which may be ga-
thered and fet down in the Table,

- Thardone, and the Horizon drawn asbetore, if you would draw the

*Parallel of 26 gr. 34 m. from the Horizon, look into the Table, and there

finding under the Title of the Parallel of 26. 34 the diftance on the South
Azimuth-line to be 7. 66, take 7 inch, 66 cent..out of a Line of Inches,
and prick them.down on the Meridian of your Plane from the Vertical
Point at V,: . oe ‘
Or if either the Vertical Point fall without your Plane, or the extent
at any time be’ too large for your Compafles, you may prick down the
diftance between che Hor'zon and the Parallel. “As here the diftance be-
tween the Vertical Point and the Parallel is 7.66, berween the Vertical
Point and the Horizon 13.44, the difference between them 5, 78 isthe

diftance from the Horizon to the Parallel, which being pricked. down

upon the Meridian, fhall give che fame Interfection as before, . And the

| Tike reafon holdeth for the pricking down the reft of cheir diftances on

their feveral Azimuths.
~ Having the Point of Interfeétion between the Azimuths and the Paral-

lel; you may joynthem all in a crooked Line, without making of Angles ;

_ the Line fo drawn thall be che Parallel required. And upon this ground tr

- followeth,
| 2 ;
Todefcribe fach Parallels on the former Planes, 48 may {hem the propors
tion of the Shadow xnto the Gnomon. .

“The proportion of a Mans Shadow unto his Heighe, or other Shadow to-

~ hisGnomon, fet perpendicular tothe Horizon, may be fhewed by Paral-

lels to the Horizon, if they be drawntoadue Altitude, which may, thus:

ie Fue

‘As the length of the Shadow,
to the length of the Gnomon =
So the Tangent of 4.5 gr
tothe Tangent of the Altitude.

V4 ‘As if it were required to find the Altitude of the Sun when the Sha-
dow of a Man hall be decupleto his Height, extend the Compafles from-
x6 wnto Bin the Line of Numbers), the fame extent will reach in the Tan-
peat.

vo - y ao ON oe ee oe ae a + “Ne x
- ne a = ~guke \ os
2 ; é Ni

96 The Proportion of Shadows, 1
Bent of 4§ gr. unto the Tangent of § gr. 42m. which fhews that when _
the Sun cometh to the Altitude of § gr.42 m. your Shadow upon a level —
Ground will be cen cimesas much as your Height. In thé fame manner —
you may find, that at-7 gr. 7 m. of Altitude your Shadow will bea duple, _
at 9 gr. 27 m. fextuple, ac 11 gr. 18 m. quintuple, at 14 gr. 2m. quadru- —
ple, ac 18 gr. 26 m. triple, at 26 gr. 33 m. double to your Height, ar 33 gr.
4I m. as 3unto 2, at 36 gr. 52m. as 2 unto 3, atZ4gr. 39m.ass unto
4, at 4g gr. equal, at st gr. 20m. as 4unto §, at 53 gr.7 m. a5 3 une
to 4, ats6 gr. 19m. as 4 unto 3, at 58 er. 2m. asjuntos, ar63gr. |
26 m. as T unto 2, eo. :
_ If then you draw a Parallel co the Horizon at § gr. 42m. another at
7 gr. 7m. and {o the reft, when the thadow of the Style fallech on the —
~ Parallel, you have the proportion, and thereby may you know. the Shadow
by the Height, and the Height by che Shadow, whereof you haveanexam-
ple tpag. 8. ROT G7 ‘a
I might here proceed to fhew the Defcription of the Circles of Pofition,
the Signs of the Zodiack in the Meridian,the Signs afcending and de(cend=
ing, with fuch other Gnomonical Conclufions: but thefe would prove —
{uperfluous to fuch as underftand the Doétrine of the Sphere ;and for
others, that which is delivered may fuffice for ordinary ufe, it being my |
intention not fo muchto explain the full ule of Shadows, (whereof I have
lately given a large example in another place) as the ufe of thefe Lines of |
Proportion, that were“not extant heretofore. ;

-

re A
ne

i. . r ee é 7 ht , i a ‘ "4 “ff;

FAPPEN DY X

i
|
|

CONCERNING THE
DEsCRIPTION and Use
| Ofafmall Portable |

UADRANI,
| “For the more eafie finding of the
HOUR ad AZIMUTH,
) AND — | |
Other Aftronomical and Geometrical Condlufions.

|
i
|
}
|
|

Of the Defcription of the Quadrant.

POR CD Aving defcribed thefe ftanding Planes, J will now fhew
g the moft of chefe Conclufions bya {mall Quadrant. This
® might be done generally for all Lacicades, by a quarter
orea| WG of the general Aftrolabe, defcribed before in the Ufe of
CY ser> che Sector ; and particularly for any one Latimude, by a
“~~ quarter of the particular Aftrolabe, there alfo defcribed 5
: which if ic be a Foot Semidiameter, may fhew the Azimuth untoa De-

“ner.
Nan t.Ulpon |

j
|
|
|

|

Sy ae = oa) nd my Pour’ ait OER ee Sy Me CP by PN Ce ‘
: : f ¥ v D \ aly 3 fi , r eS

9% Che Infeription of the General Lines,

1, Upon the Center A; and Semidiameter-A’ By defcribethe Ark BC3-
the fame Semidiameter will fet off 60 grs and the half of that will be
30 gr. which being added to the former 60 gr. will make the Ark BC :
, : Ye ij

im

| The lafeription of the General Lines. . 99
be 90 gr. the fourth part of the whole Circle, and thence comes the
name of a Quadrant. - |
2, Leaving fome little fpace for the Infcription of the Months and Days,
on the fame Center A, and Semidiameter AT, defcribe the Ark TD,
which fhall ferve for either Tropick. | |

3. Divide the Line AT in the Point E, in fuch proportion, as that

_AT being 10000, A E may be 6556, and there draw another Line E Fy.
which fhall ferve for the Equator; or AE being rog00, let ET be

253. |
4. Divide A F the Semidiameter of the Equator in the Point'G, fo as
__AF being 10000, the Line A G may be 4343: and on the Center G,
and Semidiameter GD, defcribe the Ark ED, which {hall ferve for a
| fonrth part of the Ecliptick.

|g. This part of the Ecliptick may be divided into three Signs, and each

A Table of Right Afcenfians.

30 127 $4157 48l90 Oo

ages
As the Right Afcenfion of the firft Point of w& being 27 gr..54 mm.
you may lay a Ruler to the Center A and 27 gr. 54m. in the
Quadrant BC, the Point where the Ruler croffech the Ecliptick thall

be the firft Point of &. In like manner, the Right Afcenfion of

the firft Poine of 1 being 57 gr. 48 m. if you lay a Ruler to the
Center Aand §7 g7-48 2. in the Quadranr, the Point where the Ruler
-eroffeth the Ecliptick fhall be the firft Poine of : And fo for the
_reft. But che Lines of diftingtion between Sign and Sign may be
| beft drawn from the Center G. /

None 6, The

Sien into 30 gr. by a Table of Right Afcenfions, made as followeth.

+ i ia
Z

Gre) Parts. 6, The Line ET between the Equator-and the
—- 176.) Jtopicky which I call che Line of Declination, may _
: Z be divided into 23 gr. ¢ out of this Table, For lec
2 355 | A E the Semidiameter of the Equator be 10000, the -
3 3 Hi diftance between the Equator and 10 gr. of Decli- .
| i i nation may be 1917 more ; between the Equator and’ |
EDia te tie lige er. 42813 the diftance of the Tropick from the .
4 0 as Equator 5252.
YAR zag q+ You may put inthe moft of the principal Stars. «
4 & | 1503 | beeween the Equatorand the Fropick of &, by their |
q 9 1708 | Declination from the Equator, and Right A(cenfioa
J TO} 1917 | fromthe next Equinoétial: Point. As the Declination —
11 2130 y of the Wing of Pegafus being 13 gr. 7m. the Right
12 2348 | Afcenfion 358 gr. 34:m, fromthe firft Point of 7,
13) {> 2971 | or t.gr. 26 m. fhorcof it. If you draw an occult Pa-
14 | 2799 | yvallelchrough £3 gr. 7m.-of Declination, and then
TS {_3032 ) Jay theRuler tothe Center A, and 1 gr. 26 m.in the |
| 16 3290 uadrant BC, the Point where the Ruler croffeth
17 3514 | -theParallel fhall bethePlacefor the Wing of Pegafu,
| 18 3763 | -towhich you may fet the name and the time when
19 4019 4 -hecometh to the South at-midnight ig chis manners |
| 20 | 4281 W..Peg.* 23 Ho. 54 24. And fo for the reft of thefe |
“21 | 4550 | five, or any other Stars. | |
BD 4825
23 5104.
Trop. | 5258

(Ho. MIR, Afc,[Dec. M.

Pegafus Wing *| March 3/23 5 | I O6/13 47
Ar fharns “| Odober 14/14 ©0830 o7j21 8
\Lions Heart * | eAugaft 7] 9 50)32 28/13 42
Balls Eye * | Ma 16] 4 1864 I8its 46
Vultures Heart * | Fanuary ilig 35 |

66 26/8 . 3

%. There being {pace fufficient between the Equator and the Center,.
you may there defcribe the Quadrat, and divide each of the two Sides
fartheft from the’ Center A into 100 parts3, fo thall che Quadrant be pre=.
pared generally for any Latitude.

But before you draw the particular Lines, you are to fic four Tables.
under your Latitude. : Firt,

t

ee RSS CO rere eas, CTE ESSER ee NY eee ey
| Dies. o 5 Ipe 4 TS 20 25 30
| ee renee Se et Le rer + eee

Afune GE 36/61 §4162 ol61 58/61 451/61 © 22/60

NO&ober |31 46|29 $3128 3126 16

—

To deaya Table of the Meridian Altitudes, 10%
Firft, a Table of Meridian Altitudes, for divifion of the Circle of

_ Days and Months, which imay be chus made. Confider-the Laticude
of the Place, andthe Declination of the Sun foreach Day of the Years

¥f the Latitude and Declination be alike, both North, or both Southy.
add the Declination to the Complement of the Earitude; if they be un

like, one North, andthe other South, fubcraé& the Declination from the.

Complement of the Latitude, che Remainder will be the Meridian Alti-
tude belonging unto the Day. \

Thus in our Latitude of 51 gr. 30m. Northward, whofe: Comple-.

ment is 38 gr. 30 m, the Declination upon the tenth day of Fane will be
23¢r. 30m. Northward ; wherefore ] add 23 gr. 30m. unto 38 gro.
3.0 m. the fumof both is 62 gr. for the Meridian Alcitude atthe tenth of:
Fane. The Declination upon the.tenth of December will be 23 gr. 307.
Southward, wherefore I take thefe 23 ¢r. 30m. out of 38: gr..30 ms
there will remain I5 gr. forthe Meridian Altitude at the renthof Decem=.
ler ; and in thismanner you may find the Meridian Altimde for. each
Day of the Year, and fec them down in a Table. |

6

eA Table of the Meridian Altitudes.

Months. \Gr, M.|Gr. M./Gr. M./Gr. M|Gr. M/Gr. MJGr. M,

SE

January 16 31/17 24118 ‘adélrg: 37/20 §7|22 24123 - 58
February \24 17125 $9127 45/29 35/31 29133 25
March |34 35|36 33138 32/40 30/42 27144 22146 15
April [46 37148 26]50 rr]§r S053 25154 $3156 15
May \§6 15197 29/58 35159 33/60 22161 2161 31

Faly GO 49/60 6/59 14/58 13157" 4155 48]54 24
Anguft \S4 715% 36/30 Sol49 17147 31143 41]43 26
September\43 26/41 30/30 33/37 36/35 38133 41|3r 46}

24 35/22 Solar 29
Novemb.\21 12/19 §1/18 39/17 36)16 43/16 ojrg 28
December|15§ 28 I§ Ji1y O|I§ abn I7|Iy 44 16 22,

The Table being made, you may infcribe the Months, and Days of
3 each,

SAAS yy PERE PS ee ts noe ta, Fe ae oe
nett wa: . ; |
‘

102. To fita Table for drawing and dividing of the Hivizon,
each Month into your Quadrant, in the {pace left below the Tropick?
For, lay the Ruler unto the Center Ay and 16 gr. 31 .in the Quadrant
BC, there may you draw a Line for the end of December and beginning —
of Fanuarys then laying your Ruler to the Center A, and 24 gr. 17 m.
inthe Quadrant, there draw che end of Fansary and beginning of Fea
braary, and fo thereft, which may be noted with J, F, M, A, M, J, ec,
the firft Letters of each Month, and willhere fall between 15 gr. and
62 gre.

the fecond Table which you areto fit, may ferve for the drawing and
dividing of the Horizon. For drawing of the Horizon,

Abs the Co-tangent of the Latitude,
to the Tangent of the greatef Declination:
Sothe Sine of 90 er. |
to the Sine of the Interfettion where the Horizon {hall crofsthe
Tropicks |

So in our Latitude of 51 gr. 30 m,. we thall find the Horizon tocut the
Tropick in 33 gr. gm. wherefore if you lay the Ruler tothe Center A, _
and 33 gr.gm. inthe Quadrant, thePoint where the Ruler croffeth the
Tropick fhall be the Point where the Horizon croffeth the Tropick. And
if you find a Point at Hin the Line A C, whereon fetting the Compaf-
fes, you may bring the Point at Eand this Point in the Tropick both
Into a Circle, che Point H fhall bethe Center, and the Ark fo drawn (hall —
be the Horizon. ,

Then for the divifien of thisHorizon,

As the Sine of 99 gr.
‘tothe Sine of the Latitude:
So the Tangent of the Horizon,
to the Tangent of the Ark in the Quadrant, which (hall divide
the Horizon.

So inour Latitude of §2 gr. 30m. we hall find 7 Or. $2 ms belong-

ing to 1Ogr, in the Horizon, and 1 5 fre 54m, belonging to 20 gr. And
eo che reft, asin this Table. . 8 &

.

: AT able for dividing of the Horizon: 103

3 \Gr.M. S Gr. M. > Gr.M. IGM. 2 /GrM, > Gr.M.|

9} 0 0|15 pbs 1S 30124 19/45|38 2)60153 35|75!71 5

‘0 471 112 39 Pate) 4 aha eS 54411 172 19

Ix 34] [13 27) 26 4] [40 of ss 48] [73 32

-2 21) {14 16) {26 57] {41 OO} {56 56] 174 48

3 Ble JES. Al 127 SO) 142 0] eige aha! °3

S43 SSIZO1TS 54/35/28 43150143 Ol65|59 13180177 18
4.42) |160 431 (29 37] |44 1

60°22]. 178''33

| 5 29] |17 33) [30 32] [45 3] |6r 311 I79 40
Re 6 27} |18. 22) 31 27) [46 §] |62 41] {81 5
| 7 4) [19 12, 432 22) 147 8! 162-s21 [82 27
Io] 7 s2!25 20__2)/40]33 18/55/48 11/70[65 3|85/83 37)

8 39] |20 53] [34 14! 149 14) [66 15) -!84 53

9 27 21 44) 135 10] |50 rol |67 | 86 10

10 14! {22 367 136 7| |§1 24] 168 391 (87 26

II 2| [23 27] -137 4| |52 29] {69 521 {88 43

I§|II_ $1!30'24. 19/45/38 2160]53 35!75|71 sf{ool90 oO

Wherefore.you may lay the Ruler to the Center A, and 7 Lre 52 Me

in the Quadrane BC, the Point where the Ruler croffech the Horizon

fhall be 1o.gr. in the Horizon 3. and fo for che refte: But. the Lines of .
- diftinétion between each fitth Degree will be beftdrawn from the Cen-
i ter FY.

The third Table for drawing of -the Hour-lines muft bea Table of the

~ Altitude of the Sun above the Horizon at every Hour, efpecially when he

cometh tothe Equator, the Tropicks, and fome other intermediate De-
clinations.

If the Sun be in the Equator, and fo have no Declination,

As the Sine of 90 gre
to the Co-fine of the Latitade :

So the Co-fine of the Hour fromthe Aderidlany. ?
tothe Sine of the Altitude.

Thus in our Laticude of $1 gr, 30 m. at fix Hours fromthe Meridian

the Sunwill have no Altitude, at five the Altitude will be 9 gr, 17 ! at
: four

si

4. ated
vO

$ ny

104 «0 find the Altitude of the Sun.” :
four 18 gr. 8 mw. atthree 26 gr. 7 ms, at two 32 or. 37m. atone 36 fr,
58 m. at Noon it will be 38 gr. 30. equal to the Complement of the —
Latirude.

If the Sun have Declination, the Meridian Altitude will be found as —
before, for the Table of Days and Months. | ,

If the Hour propofed be fix in the Morning or fix at Night,

As the Sine of 90 2x.
_ to the Sine of the Latitude:
. So the Sine of the Declination, ip
to the Sine of the Altitude.

Thus in our Latitude the Declination of the Sun being 23:gr. 20 a. the
Alctude will be found to be 18 gr. 11%, the Declination being 11 gr,
30 m,the Alcicude will beg gr.

If che Hour propofed be neither ewelve nor fix,

As the Co-fine of the Honr from the Meridian,
to the Sine of 90 gr.

Sethe Tangent of the Latisude,
tothe Tangent of a fourth eArk.

‘Soin our Latitude, and one Hour from the Meridian, chis fourth Ark
will be found to be § 2 gr. a8 ms. at two 55 gr. 26 m. at three 60 gr. 39m.
atfour 68 gr. 22%, and at five Hours from the Meridian 78 gr. 22 mM. /

Then confiderthe Declination of the Sun, and the Hour propofed ; if

‘the Latircude and Declination be both alike, as with usin North Latitude,

North Declination, and the Hour fall between Noon and fix, take the
Declinationout of the fourth Ark,the remainder fhall be your fifth Ark.
- But if eicher the Hour fall between fixand midnight, or the Latitude _
and Declination fhall be unlike, add the Declination unto the fourth Ark, _
_ and the fum of both hall be your fifth Ark: or if the fum fhall exceed
‘90 gr. youmay takethe Complement unto 180 gr. This fifth Ark be-
ing known,

eAs the Sine of the fourth eArk,
_ tothe Sine of the Latitude:
Ss the Co-fine of the fifth Ark,
to the Sine of the Altitude.
| Thus |

~

— Pp
| . | :

Woe /
ie

i)

||

| 23 gre 3m, of North Declination, if it thall be required to find the Al-
| titude of the Sun for feven in the Morning: here (becaufe the Latitude
_ and Declination are both alike te the Northward, and the Hour propofed
_fallech beeween Noon and fix) you may take 23 gr. 30 m. the Ark of the
Declination, out of 78 gr. 22 m. the fourth Ark belonging to the fifth

_ Hour from che Meridian, fo there will remain 54 gr. 52m, for your fifth -

2

_ Ark: Then working according tothe Canon, you thall find

As the Sine of 78 gr. 22 m. your fourth Ark;
to the Sine of 51 gr. 30 m. for the Latitude :

So the Sint of 35 gr. 8 m. the Complement of your fifth Ark, .
to the Sine of 27 pr. 17m, the Altitude required. |

_ Jf inthe fame Laticude and Declination it were required to find the
_ Altitude for five in the Morning, here the Hour falling between (x and
| Midnight, if you add 23 gr. 30m. unto 78 gr. 22%. the fum will be
401 gr. 52m. and the Complement to 180 gr. will be 78 gr. 8m, for
your fifth Ark. Wherefore, !

In Refiangulo ) DH,

UtO E Radius,

ad E MM Co-tan. Lai.
ItaO DCof, Hore.

44 DHTan,. DH.

Cufns agualis ef? P R, cwjna Compl. D Ry novis dr. arco quarts,

Con=

To fied the Altitude of the Suns rege

Thus in our Latitude of 54 gr. 30. Northward, the Sun having |

Se OLY Nee RAS ed ee ee ey aed Ce eae

106. _—-*To find the Alsitude of the Sip.

Conferatur Artus DH cum Arca Declinationis D S, ita dabitur Arcus.
HS, cujnsCompl. eff SRG prinsdr. Arcus quintus. Unde crit, 4

Ut Cofi.PR, Hoc eft, Ut Sin. DR,
iy 3 adCofi.P Z: ad SinwnE Zs:
@ ‘Ita Cofi. S Ry ; Ita Sin, HS,
| ad Cof. S Z, ad Sin. AS.

: Hine forte praftabit vocare HS Arcume quintum, ita [ecunda operatio
sn ftituetur per folos finns. |

Vel filibet (ubtrattionem finus quarth Arcus evitare, inveniatur Angulus
O HD quod fieri poteft variis modis. Nam, : .

‘ i
Pf fe ~

ae 7) ow, Wt Radives, 2, Ut Sin. DB,
ice ad Sin, Ang.O: aa Sin-O:
| TtaCof. Lat. OD, Ita Sin. D O;
ad Cofi. An. O1H D. aa Sin. A. [
jk 8. 4 Sin. D H, 4. Ui Sin. DR, |
. Spats aaTan.DO: | ad Sin. EZ, > |
Ita Radius, Ita Rad. |
adTan. Ang. Ho ad Sin, He ‘

Invento utcunque Angulo ad H, eritdn Retbangelo.H AS.

‘i Ht Sinus Refti Anguli H AS,
en vad: Sinus Arcus guinti HS:
~ Ita Sinus Anguli ad Horiz. S HA,
. Ad Sin. Solaris Altitedinis S A.

+ ste

As the Sine of 78 er.22m.

tothe Sine of $1 gr. 30 m. .
So the Co-fine of 78 er. $m. - %
to the Sine of 9 er: 32m. for the Altiinde required.

If in the fame Latitude of 51 gr. 30.2. Northward, the Sum having: |
23 gr. 30m. of South Declination, ic were required the Altitude for |
nine in the Morning : Here, becaufe the Latitude and Declination |
m2. ; are

| erieeee
me 2
or >

-areunlike, the one North, and the other South, you may add 23 gr°
30 m. the Ark of Declination, unto 60 gr. 39 m, the fourth Ark be-
_ longing to the third Hour from the Meridian ; fo fhall you have 84 gr.
- gm. for your fifth Ark. Wherefore, |

es the Sine of 60 gr. 39 m.

to the Sine of $1 gr. 30m.

| Sothe Co-fine of 84 gr. 9 m.

| tothe Sine of 5 gt. ¥5 m. for the eAltitude required.

And fo by one or other of thefe means you may find the Alcicude of

them down in {uch a Table as this.

eA Table for the eAltitade of the Sun in the beginning of each Sign
_ at-all Elours of the Day, calcnlated for 5t gr. 30m. of North
Citiindg. Me - Aes) 2): |

wy

SS ee

Sete

eos |e OO

“Sanopy

1159 43156 34/48 12136 g8l25 4ojt7 6/13 §2
2153 45|50 §5|43. 12/32 37)2r $1j13 38]10 30
$8 0/26 = *Fhrs aSoL oO. tal ys 29)
ae SUS shot So eta ee ESE ahs
18 18 9 F710 6) ods

12
II
fe)
9
8 4/36 41/34 13
7
6
5
4

TRY es aso ics Ma ee ema

Laftly, You may find what Declination the Sun hath when he rifech
> or fecceth ar any Hour. 7 ey 1

Fo find. the Alsitade of the Suto caer

‘the Sun for any Point of the Ecliptick at all Hours of the Day, and fec |

Gr, M.IGr. MJIGr. MIGr. MJGr MJGr. (Gr
62 ols 42/50 138 30/27 1l18 i8}15 oF

"
| Ooo 2 As

Sr aric Fs
FG as 1 peas ~~ oe ae

ap

9

a
ee

Nes ‘ alee hi ‘ . Tee Ce ee eed + wae
’ 4 sf a a &
} . : ae
a
<

108 The manner of deavving the Hour-lines:

“4s the Sine of 90 gr. ee eS
to the Sine of the Hour from fix :

So the Co-tangent of the Latitude,
tothe Tangent of the Declination.

And fo in the Latitude of 51 gr. 30 m. you fhall find that when the

Sun rifech, either at five in the Summer, or feven in che Winter, his De-
clination is 11 gr. 37 m. when he rifeth at four in the Sammer, or eight

m the Wincer, his Declination is 21g. 40m. which may be alfo fec |

down in the Table. . :
That done, you may there fee, chat in chis Latitude the Meridian
Altitude of the Sun inthe beginning of & is 62gr.in IE 58 gr. 42m.

nO 5Ogr. in Y 38 gr. 30 m. &c. But the beginning of Sand ¥ is ~

reprefented by the Tropick TD, drawn at 23 gr. 30m. of Declination,
and the beginning of Yand m, by the EquatorEF. If youdraw an
occult Parallel between the Equator and the Tropick, at 18 gr. 30%. of

Declination, it thall reprefent the beginning of yw, m, m, and 3, if —

you draw another occult Parallel through 20 gr. 12. of Declination,
it fhalf reprefent the beginning of TT, SU, #, and a.

_ Then you may lay a Rulerto the Center Ay and 62 gr. in theQua-
drant B C, and note the Point where itcroffeth the Tropick of $s then

move the Ruler co 58 gr. $2 m. and note where it crofleth the Parallel of

M5 then to sogr.and note where it croffech the Parallel of wy ; and again
to 38 gr. 30 m. noting where it croffeth the Equator: fo the Line drawn.
through thefe Points fhall thew the Hour of 12 in the Summer, while che

Sut isin VY, 5 Ty S) A, or Mm. Inlike manner, if you lay the —

Ruler to the Center A, and 27 gr. inthe Quadrant, and note the Point
where it croflech che Parallel ot 9¢ 5 then move it to 18 gr. 18 a. and
note where it croffeth the Parallel of aw: and again to 1§ gr. noting where
at crofleth the Tropick of yp ; the Line drawn through thefe Points fhall

fhew the Hoor of 12 inthe Winter,. while the Sun isin my TL, oy WP,

ey, and 3 fo may you draw the reft of thefe Hour-lines : enly that
of 7, from the Meridian in the Summer, and 5 inthe Winter, will crofs
the Line of Declination at 1g gr. 37 mg. and that of 8 in the Summer,

and 4.in the Winter, ac 21 fre 4.0 29,
The fourth Table for drawing of the Azimuth-lines. muft likewife be
ficred for che Altitude of the Sun above the Horizon at every Azimuth,
efpecially

rete,

To find the Sans Altitude forthe Azimuth and Latitude, 109
_ efpecially when he cometh to the Equator, the Tropicks, and fome other
mtermediace Declination. } WK De

If che Sun bein the Equator, and fo have no Declination,

As the Sine of 90 er. ;
to the Co-fine of the Azimuth from the Meridian :
So the Co-tangent of the Latitude,

to the Tangent of the Altitude at the Equator.

_ Thusin our Laticude of $1 gr. 30m, at 90 gr. from the Meridian the
Sun willhave no Altitude; at 80 gr, the Altitude will be 7 gr. 52 mm
—arzogr.itwillbe 15 gr. 30m, at 60 gr. it will be 21 gr. 41m.

If the Sun have Declination, the Meridian “Alcicude will be eafily
| found as before, for the Table for Days and Months. And for all other

_ Azimuths,

|. As the Sine of the Latitude,
ef to the Sine of the Declination :

So the Co-fine of the Altitude at the Equator,

to the Sine of a fourth Ark.

ee ,
When the Latitude and Declination are both alike in all Azimuths:
_ from the prime Vertical unto the Meridian, add this fourth Ark unto the
_ Ark of Altitude at the Equator,

When the Latitude and Declination are both alike , and. the Azi-
~muth morethan 90 gr. diftant from the Meridian, take the Altitude ac
_ the Equator out of this fourth Ark, |
~~ When the Latitude and Declination are unlike, take this fourth Ark
out of the Ark of Alticude ac the Equator, fo fhall you have the
‘Altitude of the Sun belonging to the Azimuth. !

_ Thus in our Latitude of 54 gr. 30 m. Northward, if it were
required to find the Altitude of the Sun in the Azimuth of 60 fra
from the Meridian, when the Declination is 23 gr. 30 m. North-
ward, you may find che Altitude at the Equator belonging to this
Azimuth co be 21 gr. 41 m. by the former Canon; and by this laft
Canon you may find the fourth Ark to be 28 gr. 15 m. Then be-

wf you add them

\

Alcicude required.
: : OM

caufe the Latitude aud Declination are both alike to the Northward, —
both together, you fhall have 49 gr. 56 m. for the .

nde

‘E10 ‘Th find the Suns Altit

for the

%4

Lables for the Altitude of the Sunin the

Lat. 150... Grs 00 AL.
Merid. | 10 | 20 | 30 | 40 | 50 | 60 {| 79 | 80a / go
63 30163 14462 23(60 $4158 42/55 32/$1 25146 231 7/31 22)
I60 12/59 s4tso (O17 2319S 1ST 4314-7 18]4s 40/34. 47/26 481
wist 3151 9 O 3/48 TOl4s 23/41 3436 35130 39/23 r2|I5 5}

¥140 ol39 34/38 15:36 0j32 44128 20|22 45\60° o 8 17fO of,

¥/28 30/28 026 27/23 solz0 sity 6; 8.§2] I 30] 6 38
ait 48\T9 I4j17 31/14. 37/10 27) 4 57 I 43h 9 40/18 13
Wi16 30\I4 $4114 7/II 6| 6 46, I 95! 6 5814. 2(22 43
r Lat, §t. Gr. ;

G 62 30162 1462 22159 s4l37 40194 35/59 27/45 8/38 33/30 53)
159 12158 5457 59/56 23154 Of0 43]46 22i4r 51/34 6/26 23]
wIIO 3050 749. 3147 11/44 25140 4035 47/29 48i22 43/14 52
W399 0138 34137 16135. 3134 49127 30/22 2\t5 291 8 of 0
%27 30/27 125 20(22 595| 9 13|14 20] 8 17] 1 10] 6 43
mg 18° 481/18 14.16 33}13 43| 9 38) 4 17} 2 18] 9 $3118 6
wly 30}14 $413 IC}10 121 § §8] O 25] 6 23/14. 1022 33

, ' Lat. Ԥ2 Gr, |
@lS1 30/91 14|60 22|58 52/56 38]53 33149 20144 14/37 58/30 24]
T}y8 12|58 54/56 28)56 22153 O149 43/44 25/40 0/33 2826 OF
wI49 30149 9148 3146 11\43 26/39 44134 S8]20° 622 15'T4. 40
¥158:.19l47 95136 37/94. 1130) S4k6 40)21.20)14 971 7 44) oa
%126 3026 1/24 31/22 Of18 22]13 26]17 42] 0 48) 6 46
eeli77 48/17 6s 3612 48] 8 40l 3 3712 45l10 618 ©
vpjt4 30114 JO|I2 121 9 18} ¥ 10} O 13) 6 4914 1922 30

ae every tenth Azinsuth.

Azimuth and Latituie,
OM 90. 60, |
ME Comp. Lat.
O A Com, Azim.
AB Alt. Equa.

meat fy

EZ Las.

Z BCom. AB.
DS Deelis.
SB Are. 4,

beginning of each Sign for

4. °, Oe
S| cee

South part of the Meridian. The like reafon holdeth for the re(t of thefe
_ Altieudes, which may be gathered, and fet down in a Table, f

-_ Laftly, when the Sun rilech or fecteth upon any Azimuth, to. find his —

- Declination.

As the Sine of 90 gr.
ag tothe Cofine of the Latitudes
50 the Co-fine of the Aximnth frows the Meridian,
to the Sine of the Declination. ee

nn a a a EES,

eA Table for the Altitude of the Sun in the beginning of each Sign for

every tenth Azimuth, in 51 gr, 30 m. of North Latitude.

S$ if RS MV oI mies zl

i} murhs. Gr, M.|Gr, M.|Grs M.|Gr M.iGr. M/Gr, M./Gr. M.

62 0158. 42|50 0,38 3027 jo18 1845 Ol,
JOr 43158 24/40 38/38 4126 30117. gsli4. 25],

{60 $1157 28148 33/36 46lay .o0]16 gii2 41

$4 3/§0 12/40 11127 $]13, §8) 3 g7j-0... 6
49 56/45 §093|35 23/21 4a].8 0 fess

M44 40/40 29/29 27)1§..43),1 To

38 11133 46.21 291-7 fab.

30 38/26- 10]14 25]-0. ©

| [22 27|78 2.6 ay. 6 12
tonig tle’ se iA 8g
GilFH B2i90N bun whsiend, hice nl8. <4 SU

~ And chusin our Latimdeof §1 gr. 300. when the Azimuth is 80 gr.

The Infeription of the Azimaths, - wit
Af the Declination had been 23 gr. 30 m. tothe Southward » you
_ fhouldithen have taken thisfourth Ark out of the Ark at the Equator;
_ which becanfe it cannot here be done, it isa fign chat the Sun is not then
above the Horizon: Barif youtakeche Ark ac che Equator out of this.
- fourth Ark, you fhallhave 6 gr. 34m. forthe Altitude of the Sun when
“heisinthe Azimuth of 60 gr. from che North, and 120 gr. from the. —

ap Same |

2 $2155. 52140 40/34 34)22 27113 25) 9 45}:
E 20,53 29143 $9131 21518 48) 9 14,5 341,

E
ea

wee Le ee ee Se eT J. A Sta «oe
eh SU
. ' a) YS. one?

Tit The Infeviption of the Azinuths

from the Meridian, the Declination will be found tobe 6 gre 13 m. if |
che Azimuth be 70 gr. the Declination willbe found 12 gr.18 m.it 60 gr.

then 18 gr. 8a. And fo for the reft, which may be alfo fer down in the
Table. ‘ig

where it croflech che Tropick of % 5 then move the Ruler to 26 gr. 10 m.

and note where it croffech the Parallel of [73 then to 14 gr. 4§ m. and —
note where it croffech the Parallel of y thento © gr.o mand you thalf

find it to crofsthe Equator in the Point F : fo a Line drawn through thefe

Points (hall (hew the Azimuth belonging to Eaft and Weft. The like —

rea(cn holdeth for all the reft. |

Thefe Lines being thus drawn, if you fer two Sights upon the Line —

That done, if you would draw the Line of Eaft or Weft, which- ig
gogr. from the Meridian, lay the Ruler tothe Center A, and 30 gre *
38 m. numbred*in the Quadrant from C toward B, and note che, Point —

AC, and hang a Thred and Plummet on the Center A, with a Bead —

upon the Thred, the Fore-fide of the Quadrant fhall be fully finithed.

On the Back-fide of the Quadrant you may place the Noéturnal de=.__

{cribed-before in the Ufe of the Seftor, which confifteth oF two parts. |

|
|

The oneis'an Hour-Plane, divided equally: according to the 24 Hours _
of the Day, and each hour into Quarters, or Minutes, as the Plane will

bear... The Center reprefents the North Pole.; the Line drawn through |
the Center from XII to XII ftandsfor the Meridian, and thelower XIT

f{tands for the Hour of XII at midnight. .

The other part isa Rundle for fuch Stars as are near the North Pole,
together with che rwelve Months, and the Days of each Month, fitted to
the Right Afcenfion of the Sun and Stars, in this manner,

*
ry
a

Firft, confider where the Sun will be at che beginning of the 5, 10, |
15) 20, 2§, 30, and, if you will, every day of each Month, and find
the Right Afcenfion belonging to the place of the Sua, as I thewed.

before. | | ; » |
For example: The Sun at midnight, the laft of December,’ or begin-.

ning of Fansary, will be communibus annis about 20 gr. 40 m. of Wy,

Right Afcenfion is 324 gr. 35 m, and fo thereft, which may be fet down
in \a Table. ! 3

%
/

‘That done, confider the Longitude and Latitude of the Stars and,
thereby fiad their Right A fcenfion and Declination as I thewed before,and

fer them down ina Table. Thefe Tables thus made, let the uppermoft
\pare

(

whofe Right Afcenfion is 292 gr. 20 m. At midnight, che laft of fanu-

ary, or beginning of Febraary, he will beabout 22 gr. 12 1. of ws, whofe! —
. |

3
5

Perret om
£ "aa

> the Rundle be ma

Then lay the Cencer of this Rundle

ed into 360 gr, and by the
Center and 292 gr 20 ms. in that Circle, draw a Line for the

beginning of
| Fansary: Inlike manner, by the Center and 324 gr.35 m. draw a Line

beginning of February; and fo the reft of the
_ Daysof each Month.

| For the Infcription of the Stars, let one of the Lines from the Center,
as thac at the beginning of Faly, or rather let a movable Index be divided
| from the Center toward the inward Circle of the Months into 40 gr.
| more or lefsy which may be done for {peed equally, but for exa@nefs in
- fach manner as the Semidiameter of the General Aftrolabe was divided
before in the Ulfe of the SeSor, So laying the Index to the Righe Afcen-
“ fion in the outward Circle, you may prick down the Stars by their De-
clination inthe Index, < | ia

For example: If the Right Afcenfion of the Pole-ftar be 6 gr, 28 wy.
and his Declination 84 Ere 20 m. having fet che Center of the Index both
to the Center of the Rundleand of the other Circle, turn the Index to
6 gr.28 m, inthat outward Circle, and prick down the Star by 87 gr.
20m. inthe edge of the Index, that is, at the difance of 2 ¢r, 40m,
from the Pole. The like rea(on holdeth for che reft of the Stars, which
"may be diftinguithed according to their Magnitudes, and then be reduced
into their Forms, as in theExample. So the

m oa

Quadrant will be fitted both
| for Day and N ight. :
a CHAP. Il.

Of the Ufe of the Quadrant, in taking the Altitude of the Sun,
ia - Moon, and Stars.

il gogr. and here numbred by
| Degree being {ubdivided into 4.
Lift up the Center of the Quadrant fo as the Thred with che Plummet
“May play eafily by the Side of ir, and the Sun-beams may pafs through
both the Sights; fo fhall the Degrees cut by the Thred thew what is the
| Adtitude at the time of obfervation, as may appear by this Example.

a , p

Pp Upon

| - He Quadrant is the fourth part of a Circle divided equally into
| 10) 20, 30, @e. unto go gr. each

S| Theis. 1x3

ABR.
es

a a

apes
ar +
pee

~ ming of #,there muft be more than 90gr,. allowed to the Right Afcenfi- —

hw Nite ‘eos we a” ee 2 oe

114 — The Vie of the Ecliptick, =
Upon the 14 day of April about Noon,the Sun-beams paffing throt
both the Sights, che Thred fell upon §1 gr. 20 mm. and this was ther es
Meridian Alcitude of the Sun for: that day, in this our Latitude.of s1gr,
30 m, for which this Quadrant: was made. : | aa
Again, cowards three of the Clock in the afrernoon the Thred fellupon,
38 gr. 40 m. and fuch was the Suns Altitude at that time. <a,

€ H A ite Lit..
Of th ECLIPTICK.

1, The Place of ‘the Sun being given, to find bis Right Afcenfions.

5 Gee Ecliptick is here reprefented by the Ark figured with che Chara:
Gters of che 12 Signs, 7° By I) Cc. each Sign being divided un-. |
equally into 3@ gr. and they are to be reckoned fromthe Character of the
ign. Mpa ie |
‘Let the Thred be laid on the place of the Sunin the Ecliptick, and the |
Degrees which it cuttethin the Quadrant fhallbe the Right Afcenfion re-
quired. s
Asif the place of the Sun given be the fourth Degree of 11, the Thred }
laidon this Degree thall cut 62 gr. in the Quadrant, which isthe Right
Afcenfion required. ; 4
But if. the place of the Sun-given bemore than go gr. from the-begin=

at

on ; for this Tnftrument is but a Quadrant. And fo if the Sun bein 26 So.
of g, you thall find the Thred to fall in the fame place, and yet the Right
Af{cenfion to be 118 gr.

2. The Right Afcenfion of. the Sun being. given, to find his Place in the:
Eclipticke | ° |

~ Let the Thred be laid on the Right Afcenfion in the Quadrant, -andit —
Shall crofs the place of the Sun in the Ecliptick, as‘may appear inthe for=
mex Example. — beng | |

se Oe ME RN NT | _ Rua ence a
Po, alos aay

The Ufe of the Line of Declination’ =n

Of the Line of Declination:

x. The Place of the Swe being given, to find his Declination:

YF He Line of Declination is here drawn from the Center to the begia-
_ && ning of the Quadrant, and divided from the beginning of ¥
downward into 23 gr. 30 m. a ye :
Let the Thred be laid, and che Bead fet onthe Place of the Sun in che

_ Ecliptick 5 chen move the Thred to the Line of Declination, and there
| the Bead thall fall upon the Degrees of the Declination required,

| _ Asif the place of che Sun given be thefourth Degree of 11, the Bead
| firft fec to chis place, and then moved to the Line of Declination, thall
| there thew the Declination of the Sun at that timeto be 21 gr, from the
Equator.

| _ 2. The Declination of the Sun being given, to find bis place in the
| Ecliptick.

Let the Thred aud Bead be firft laid to the Declination, and then mo-
| ved to the Ecliptick. | ;
|- As if the Declination be 23 gr. the Bead firft fet to this Declination,
and chen moved to the Ecliptick, thall chere thew che fourth of g, the
fourth of 3, the 26 of %, and the 26 of vs and which of thefe four is _
the place of the Sun, may appear by che Quarter of the Year.

CHAP. V.
Of the Civcle of Adonths and Days.

\ je Circle is here reprefented by che Ark figured'with thefe Letters,
| J, F, M, A, M, @c. fignifying che Months Fansary, Eebyaary,
March, April, &c. each Month being divided unequally, according to
__ the number of the Days that are therein.

i «
{

Ppp A Table

| ge be i og gh Ss ee
216 The Ufe of the Circle of Months and Days. .

A Table for the Infcription of the Months in the Nottarnal, |

RPG PAE TS MEET MIST, fi
Dies. fe) 5 IO | TS 20 25 30
ee

Gr. M. Gr. M Gr, M. Gr, M, r

308 21/313 30|318336|323 36]

Menf. {Gr. M,/Gr, M./Gr. M.
fannary |292 20|297 46/303 7
Pebruary!324° 35/329 28/334 16/339 11343 421348 21 “fl

arch |351 171355 §2] 0 26 4 5819 30l 14 af 2g 34h
epril | 19 301-24 4) 28 421 33 231 38 5] 42 52] 47 426
May | 47 42) 52 35] $7 32] 62 34] 67 30 72 4s] 77 52h
June | 78:95) 84 5) 89 17/ 94 28] 99. 39/104 48lr00 55}
July |109 S3/T1§ O]12@ O124. 58!120 541134 451139. 30 4
Auguft \140 27\145 91149 48]154 25|159. 0163 32/168 of
Septemb.{168 §7|173 26/177 36]182 26}186 §6|t91 cl me |

Ottober [196 §|200 45/205 25}210 12/215 3/220 © 225 oluml
|Novemb.|226 2/231 10|/236 23/241 40 247 2/252 30/258 2h

Decemb.|258 2|263 351269 8)274 42/280 16 285 46|2091 1% :
NF, Se R ARON TN cM |

|
ms |

iB The Day of the Month being Given, to find the Altitude of the saa

at ‘Noon, }
Let the Thred be laid tothe Day of che Month, and the Degrees ja
it cucteth in the Quadrant thall be the Meridian Altitude required. 4

As if the Day given be the rs of A¢ay, the Thred laid on this day |

thall cut 59:gr. 30 w, in the Quadrant, which isthe Meridian Altitude
required. 8

2. The Meridian Altitude being iven, to find the Day of the Afonth. _

The Thred being fer to the Meridian Altitude, doth-alfo fall on the day
of the Month, | |
. Asif the Altieude at Noon be 59 gr, 30 m. the Thred being fetto his
Altitude, doth fall on the 1 5 of May, and the 9 of July; and whichof
thefe ewo is the crue day, may be known by the quarter of the year, or by,
another days Obfervation. For if the Altitude prove greater, the Thred

_ willfallonthe 16 day of Afay, and the 8 of Fuly 5 orif it provelefter,
5 a . ‘i ih: q the a

ion is Fully anfwered,
CHAP. VI.
Of the Hour-lines.

r “Hat Ark which ts drawn upon the Center of the Quadrant by the
_ beginning of Declination, doth here reprefent the Equator: That

=== —

*Circle of Months and Days, reprefenteth the Tropicks: Thofe Lines
which aré between the Equator and the Tropicks, being undivided, and
numbred at the Equator by 6,7,8,9, 10,11, 12; ac the T ropick by:
“T, 2, 2,4, &e. do reprefent the Hour-circles: That which is drawn from
42inche Equator co the middle of Fune, reprefenteth the Hour of 12 at
Noon in the Summers and thofe which are drawn with it to the right
hand, are for the Hours of the Day in che Summer, and the Hours of the:

\thofe which are drawn with itto the lefthand, are for tle Hours of the
Day in the Winter, and the Hours of the Night in che Summer 5; and of
‘both thefe, that which isdrawn from 11 to 1 fervesfor 11 inthe fore-
‘Noon, and « intheafternoon; that which is drawn from 10 to 2, ferves
for soin che forenoon, and 2in the afternoon: for the Sun on the fame
|
|

‘day is about the fae height two Hours before Noon, .as two Hours after:
‘Noon. The like reafon holdech for the reft of the Hours,

4, The Day of the Month, or the Height at Noon being known, to find
| the Place of the Sunin the Ecliptick.

The Thred being laid to the day of the Month, or the height at Noon,
(for one gives the other by the former Propofition) mark where it croflech
the Hour of 1.2, and fet the Bead to that InterfeGtion ;.then move the
bs cill che Bead fall on the Ecliptick, and it fhall fafl onthe p'ace of
‘the Sun.

Asif the day given be the 15 of AZay, or the Meridian Altitude 59 gr. °

30 m. lay the Thred accordingly, and put the Bead to che InterfeGion of
‘the Thred with the Hour of 125 then move the Thred cill the Bead fall:
Ba os

\*

r

ano or ee
ed will fall on the 14 of Aday, and the 10 of Fly; whereby the

_ Ark whichis drawn by 23-gr. 30 m, of Declination, and is next abovesthe-

Nightia the Winter: That which is drawn from.#2in the Equator, to-
the middle of December, reprefenteth the Hour of 12 inthe Winter; and

-on the Edliptick, and ir thall there thew the fourth of Tr, the fourt 1

- Noon’50 gr.o m. at 1t in the morning 48 gr. 12%. at1o but 43¢r.
“12 mat 9 but 36 gr. at 8 but.27 gr.30 m. at7 but 8 gr. 18 m.at 6 but

1180 i(ié‘«ér hE fe of the Lousy’ =Uinerse
wf, the 26 0f , and the26 of yp; and which of thele is the place c
the Sun, may ‘appear by the Quarter of the Year, or another days obfer=_

vatlone 3

2. The place of the San inthe Ecliptick being known, to find the Day of
the Month. i ae. |
Let the Thred-and Bead be firft laid on che place of the Sun im che Ecli-
ptick, and then moved to the Line of 12. | ie
+ -Ass if the place of the Sungiven be the foarth of 11, the Bead being —
laid co this Degree, and then moved to the Hour of 12 inthe Summer, .|
the Thred will fall onthe 25 day of Afay, and the 9 of Falys orif icbe |
moved to the Heur of 12 in the Winter, che Thred will fall on che 6 of |
January, and the 16 of November : which of thefe is the day of the
Month required, may appear by the Quarter of the Year. |
‘In thisand the former Propofitions you have two ways to re@tifie the
Bead, by the place of che Sun, and by the day of the Month: the better
way is by the place of the Sun.5, for in the other, the Leap- year may breed
{ome {mall difference. |
There is yet a third way : For the Seamen having a Table. for che De-
clination on.each day of the year, .may fer the Bead thereto in the Line of |
Declination. | |
3. Fhe Hour of the Day being given, to find the Altitude of the San
above the Horizon. 4

The Bead being {er for the cime by either, of the three ways, lec che
Thred be moved from the Hour of 42 toward the Line of Declination, —
tillche Bead fall on che Hour given ; and the Degrees which ic cuts in
the Quadrant, fhall thew the Altitude of the Sun ac thac time. |

‘Asif thetime given be the 16 of April, the Sun being then in the |
beginning of y,-the Bead being rectified, you thall find the Height ac

pgr.at § ic meeteth wich che Line of Declination, and hath no Altitude ©
at all, and therefore you may think it did rife much about that Hour. |
Then it you move the Thred again from the Line of Declination to-
‘ward the Hour of 22, you fhall find that the Sun is 8 gr. 33 2. below
SBN | the»

7

ee Ae ye ee

' the Horizon ar 4 in the morning, and near 16 gr, at 3, and 21 gr. g1 ms:
and 25 gr.4.0 m. atl, and 27 greatmidnight. ~

7 ‘a 4. The Altitude of the Sun being given, to find the Hour of the Day.

| The Alticude being obferved as before, lec the Bead be fet for the
\ time, then bring the Thred to che Altitude, fo the Bead thall thew the
|. Hour of the day. |

| Asif the 10 of April, Having fet the Bead forthe time, you thall find
by the Quadrant the Altitude to be 36 gr. the Bead at the fame time will
- fall upon che Hour-line of 9 and 35 wherefore the Hour is 9 in the fore-
noon, ov 3 intheafternoon. If the Altitude be: near 40 gr, you thalf
find the Bead at the fame time to fall half way between the Hour-line of
g and 3, and the Hour-line of to and 25 wherefore ic muft be either
half an Hour paft 9 in the morning, or half an Hour paft 2 in the after-
noon ; and-which of thefe is the tue time of the day, maybe foon
_ known by a fecond Obfervation : for if the Sunnife higher, ic isthe fore.
noon; if it become lower, it is the afternoon.

below the Horizon.

' he Sun is always fo much below the Horizon atany Hour of ‘the
Night, ashis oppofite Point is above the Horizon at the like Hour cf ‘the
Day; and therefore the Bead being fet, if. the queftion be made of any
‘Hour of che Nightin'the Summer, then move it to the like Hour of: the
Day in the Winter ;-if of any Hour of the Night in Winter, then-move
| it to the like Hour of the Day in Summer: fo the Degrees which the
Thred cutteth in the Quadrant fhall (hew how much the-Sus is. below
_ the Horizon at that time.

Asif it be required to know how much the Sun is below the Horizon
the 10 of - April at 4 of the Clock in the Morning, the Bead being fet to
his place according to the time in the Summer-hours, bring it to 4 .of the
© Clock in the,afternoon in the Winter-hours, and fo thall you find the

the Sun below the Horizon at that time.

The Vie of the Hour-lises. 119

5. The Hour of the Night being given, to find how much the Sun is

| Thredto cut 8-gr. and about 30 m. in the Quadrant; and fo muchis-

6, The:

See

» to 18 gr. in the Quadrant, fo thall the Bead fall among the Winter= _

BS han

| BA , Bs cat a Seb
6. The Depreffion of the Sun (uppofed, to give the Hour of the Night
with ws, or the Hour of the Day to oar Antipodes, © ae

iz Saga |

Flere alfo, becaufe theSun is fo much above the Horizon at all Hours of |
the day, as his oppofite pointis below the Horizon at the like Hour of the |
Night; therefore firft {ec the Bead according to the time, then bring the |
Thred to the Degree of the Suns Depreffion below the Horizon, fo thall
the Bead fall on the contrary Hour-lines, and there thew the Hour of the
Night in regard of us, which is thelike Hour of the Day toour Antipodes,
As if the 10 of 4pri/, the Sun being then in the beginning of y,.
and by fuppofition 8 gr. 30%, below the Horizon in the Eaft; it be re-_
"quired co know whactime of the Night ic iss firft, fecche Bead accord-_
ing to the Day ia the Summer-hours, then bring the Thred to $ LV. 30 14,
in the Quadrant, fo fhall che Bead fall among che Winter-hours on the.
Line of 4 of the Clock in the afternoon: wherefore to our Antipodes it is
4 of the Clock in their afternoon, and -to us itis then 4 of the Clock in.
‘the morning. | |
q- The time of the Year, or the place of the Sun being given, to find
the beginning of Day-break, and end of Twilight. |

This Propofition differeth little from the former: for the Day is'faid_
to begin to break when the Sun cometh to be but 18 gr. below our Ho-
rizon in the Eaft; and Twilight to end, when it is gotten 18 gr. below
the Horizon in the Weft’: Wherefore let the Bead be fet for the time, and
then bring the Thred to 18 gr. in the Quadrant, fo thallthe Bead fallo Ma
the contrary Hour-lines, and there thew the Hour of Twilight, as before,

Soif ic be required to know at what time che Day begins to break on
the 10 of April, che Sun being chen in the beginning of wy; firft,fecche
Bead according to the time in the Sammer-hours,and then bring theThred —

thours a liccle more than a quarter. before 3 in the morning; and chat ts
the time when che Day beginsto break upon the 10 of April.

CHAP}

+ eee as eee * eed et

' WAS RA NES se eae Re

.

ad

‘Nas eh The Dfe of the Horizon. ar

CHAP. VII.
Of the Horlzon,

°

_ and numbred by 10,20, 30, 40, Gc.

Xo The Day of the Months or the Place of the San being known, to find
the Amplitude of the Suns Rifing and Setting.

Let the Bead re&tified for the time be brought to the Horizon,and there

_ it thall thew the Amplitude required, :

| Asif the day given be the 15 of A¢zy, the Sun being inthe fourth De-

gree of I, the Bead rectified and brought to the Horizon, fhall there fall

on 35 gr. 8m. fuch isthe Amplitude of che Suns Rifing from. the Eaft,
and of his fecting from the Weft; which Amplitude is always North

_when the Sun is in the Northern Signs, and when he is in the Southern
Signs always Souchward, :

4, The Day of the Atonth, or the Place of the Sun being given, to find
the Afcenfional Difference. Lg

i Lethe Bead rectified for the cime be brought to the Horizon, fo the
Degrees cut by the Thred inthe Quadrant thall thew the difference of Afe

_cenfions, ‘apr |

__ Asif the day given be the 15 of Afay, the Sun being in the fourth De-
gree of m1, let che Bead be rectified and broughc to the Horizon fo thall

the Thred in che Quadrant thew the A{cenfional difference to be 28 gr.

and about 50 ms. | )
___Elponthe Afcenfional difference depends this Corollary.

To find the Howr of the Rifing and Setting of the San, and thereby the
length of the Day and Night. :

The time of the Suns Rifing may be gueffed at by the 3 of the laf
Qaq and

| Ae Horizon is here reprefented by the Ark drawn from the beginning
y of Declination towards the end of February, divided unequally,

“Chapters buthere by the AfcenGonal difference it may be better found, —

ia oe

ime. S TORR SEGA ROLE! Tk SCOR On ia eh er
h, he oy Pg ke tg A ea TO
z22 To find the Hour of the Night by the stars.
and that to a minute of time. For if the Afcenfional Difference be
converted intotime, allowing an Hour for 15 gr. and 4 Minutes of an
Hour for each Degree, it fheweth how long the Sun rifeth before fix of the —
Clock in the Summer, and after fix in the Winter. fei
As if. the day given be pe 1§ of Afay,. the Sun being inthe fourth of ©
HL, and his Afcenfional Difference found as before 28 gr. $0 m. this con-
verced into time,makech 1 Ao, and fomewhat morethan 55 . of an Hours
wherefore the Sun ar that rime,in regard it was Summer,rofe 1 Ao. and full _
§§ m. before 6 of theclock 5 and fo having the quancity of the Semidiur=
nal Ark, the length of the Day and Night need not be unknown,

ee

CHAP. VIII.
Of the Five Stars.

[ Might have put in more Stars, but thefe may fuffice for the finding of
the Hour of the Night ac all times of the Year: And firft I make
~ choice of Ala Pegafi, a Star in the extremity of the Wing of Pe es |
regard it wants but 6 minutes of timeof che beginningof ¥ ; but becaufe |
it is but of the fecond magnitude, and not always to be feen, IT made |
choice of four more, one for each quarter of the Ecliptick, as Oculus &
The Bulls Eye, whofe Right Afcénfion converted into time, is 4 fo. 15m, |
then of Cor §., The Lions Heart, whofe Right Afcenfion is 9h. 48 a. |
next of Arélarvs, whole Right Afcenfion is 13 ho. §8 a. and Wily of
Aquila, or The Vultures Heart, whofe Right Afcenfion is19 bo. 33 mz.
Thefe five Stars have all of them Northern Declination: and if ate :
others, fome of thefe will be feen at all times of the Year. The ufeof - |
them is, oa BM

,
—

The Altitude of any of thefe five Stars being known, to find the Hour of
‘the Night. Sa

Firft, put the Bead to the Star which you intend to obferve, take his
Altitude,and find how many Hourshe is from the Meridian by the fourth
Prop. of the fixch Chap. then out of the Right Afcenfion of the Star, rake
the Right Afcenfion of the Sun converted into Hours, and mark the dif
ference 3. for this difference being added to the obferved Hour of the Star

_ from the Meridian, thall thew ow many Hours the Sun is gone fromthe _

Meridian,which js in effeCtthe Hour of the Night,. © So

a The Ge of the Azimuth lines. 123
Asif the 5. of Afzy, the Sun being in the fourth of 11, I thould fet the
Bead to Aréturus, and obferving his Altitude, thould find him to be in the

‘Welt about 52 gr. high, and the Bead to: fall on the Hour-line of 2 after-
~ noon, the Hour would be it 4o. 50 a. paft-noon,or Yo’m. hort of mid-

night.
For, 62 gr. the Right Afcenfion of the Sun, converted into time, makes

| 40.8 m. which if we take our of 13 Ao, 58m. the Right Afcenfion of

_ Ariturms, che difference will be g ho. 50 m, and this being added to 2 ho.
_ the obferved diftance of eAréturus from the Meridian, thews the Hour of

the Night tobe 1140. 50. Another Example will make all more plain,
_ If the gof Faly, the Sun being then in 26 gr. of &, I fhould fet the

Bead of Ocalus g, and obferving his Alticude, thould find him to be in

the Eaft about 12 gr. high, and the Bead to fall on the Hour-line of 6

before Noon, which is18 o. paft the Meridian, the Hour of the Night

would be better thana quarter paft 2 of the clock in the morning.
For, 118 gr. the Right Afcenfion of the Sun » converted into time,
make 7 he. §2 #. this taken out of 4 bo. t§ min. the Right Afcen-

“fion of Oculus, adding a whole Circle, (for otherwife there could be

no fubtraétion) the difference will be 20 he. 23. and this being added
to 18 ho. which was the obferved diftance of Oculus w from the Meridi-

an, {hews that the Sun (abating 24 ho. for the whole Circle) is 1.40.23 mo
_ pat che Meridian, and therefore 23 mz. paft 2 of the clock inthe morning.

If the Noéurnal be placed on the back fide of the Quadrant, you may

avoid this Equation of Right Afcenfions. For knowing the rime of the

Year when the Scar will bein the South at midnight, you may bring that
time to the Hour obferved, then will the Day of the Month wherein you
made the Obfervacion point at the Hour of the Night required,

Asin the firft Example, where, on the 15 of May, the Bead fet to Are

Gurus fell on che Hour-line of 2 afternoon, becaufe Arurus will be in the

South the 14 of Ofgber complete at midnight, you may place the 14. of

" Offeber at the Hour of 2, fo the1g of Afay will point to 2% ho. 50 m.

_ Inthe fecond Example, where the 9 of Fuly the Bead fer to che Balls
Eye fell on the Hour-line of 6 before Noon, becaule the Bulis Eye will be
in the South the 16 of AZay complete at midnight, you may turn the
16 of May to the Hour of 6, and fo you thall-find the 9 of July to point —

2b0, 23 ms, as before,

—§ Qqq2 CHAP.

.

1240s THOU Of the Aximmuth-Lingss :

CHAP. IX.
Of the eAzimuth-lines,

Mi Lines which are drawn becweenmthe Equator and the Tropicks, _
on that fide of the Quadrant which is neareft unto the Sights, and |

are numbred by 10, 20, 30, @c.do reprefent the Azimuths; the utters —
moft cowards the left hand reprefenteth the Meridian 5 that which isnum-

bred with ro, the tenth Azimuth from the Meridian ; and that which is

numbred with 20, the twentieth: and fo the reft, Thofe Lines which

are drawn from the Equator to the left hand, do fhew the Azimuth in the

Summer 5 and thofe other co the right hand do fhew the fame inthe Wir- ,

tere The Ufe of them is:

g. Tne Azimuth whereon the San beareth from us being known, to fink }

the Altitude of the Sun above the Horizon.

Firft, fer the Bead be fee for che time, asin the former Chapter; chen

move the Thred until the Bead fall on che Azinwath: fo the Degrees which |

the Thred cuccetlein the Quadrant hall thew che Alticude of the Sun at
that time. Where you are to obferve, That feeing the Azimuths are drawn
on the right fide of the Quadrant, you are alfo to begin to number the

Degrees of the Suns Altitude from the right hand toward the lefe: As if
the Sights had been fet onthe Line A B, and you had curned your right

hand towards the Sun in obferving of his Altirude, contrary co our pras

_ ice in the former Chapter.

Asif che time given were the 2 of dugn/?, when the Sun hath about

s.gr. of North Declination, you may {ec the Bead for the time, fo you
fhall.find the Height ac Noon, when the Sun isin the South, to be §3gr.
30 m. when heis 10 gr, fromthe South 53 gr.10 #. when 20 gr. then
about §2gr..8 m. when 30 gr. then 50 gr.20 m%. when 40 gr, then 47 gr.

48 m. when 50¢r. then 44 gr. 12m, when 60 gr. then 39 gr.35 When
-ogr. then 33 gr.50 m. when 80 gr. then 27.gr. when heisin the Eaftor
Welt 90 gr. from the Meridian, then is the height near 1 9gr.20m. when —
he comes co be 100 gr. chen 4% gr, 15 ws. when r1ogr. chen 3 gre 20m.
and before he cometh co the Azimuth of 120 gr. hehath no Altitude. For —
the Sun having ts gr. of North Declination, will rife and fet ac rr4.grs

34 m, fromthe Meridian. 2,. The:

ie ~The Ufeof the Quadvat. m 125°

2, The eAltitnde of the Sun being given, to find on what Azimuth he
—. beareth from ns /

| . j
Let the Bead be fer forthe time, and the Altitude obferved as before s
then bring che Thred to the Complement of that Altitude, fo the Bead
-fhall thew che Azimuth required. .
As if the fecond of Auga/?, having fet the Bead for the time, you fhalf
findtke Alcicude of the Sun to be 19 gr. 20 m. remove the Thred unto -
-Jogr: 40m. the Complement of the Alcitude; or, whiclrisall one, to
19 gr. 20 m. from the right hand toward the left, andthe Bead will falt
on the Line of 90 gr. from the Meridian ;. and therefore the Point where-
‘on the Sun bearech from us is one of thefe cwo, either due Eaft, or due
“Weft: And which of thefe isthe true Point of the Compas, may be
foon known by a fecond Obfervation; forif the Sunrife higher, icisthe
forenoon ; if i be lower, itis the afternoon.

_ By knowing the Azimuth or Point of the Compafs whereon the Sun:
‘bearech from: us, it iseafiecofind, |

| eA Meridian Line, and thereby
The C oa fing of the Countrey,
The Site of a Building;
. The V ariation.of the Compafs..

__Asif the fecond of Auguf in the afternoon I fhould find by the Heighe
of the Sun thar he bears trom me 60 gr, from the Meridian toward the
Weft; chen there being 90 gr. belonging to-each quarter, the Weft will
be 3@ gr. tothe right hand; che Eaft is oppofite co the Weft, the North.
and South lie equally between them.

CHAP. X,.
Of the Oadrat..

He Quadrat hath two Sides divided ; the other two Sides next the
SE Cegrer may be fuppofed to be divided each of them into 100 equal
parts: That whichis nexe the Horizontal line contains the parts of Right
‘Shadow ; the other nexe che Sights, the parts of Contrary Shadow. The
‘le of the Quadrar is, we |

| T. Anyi

te ee eee
fora oe. +
;

26 THUR UF the Onadratd?

-the Diftance between the Place and the Eye. “a

the Diftances If ic fallon 25, ic isa quarter of the Diftance ; if on 755

Me he ah AD Seat 7 a. AE AOS SE eC RLCOR ET CRD
Se PA en Ges eic! Wines dated gi yk

v

1. eAny Point being given, to find whether it be level with the Eye,
Lift up the Center of the Quadrant, fo asthe Thred with the Plum

met may play eafily by the Side of ic: then look through the Sights to
the Place given; for now if the Thred thall fall on A B the Horizontal.
‘line, then is the Place given level with the Eye: But if it thal fall within
the {aid Line on any of the Divifions, then it ishigher if without, them

it islower than the level of the Eye.

Ln
2. To find an Height above the Level of the Eye, ora Diftance at one.
Obfervation. | |

Leok through the Sights to the Place, going nearer or farther from it,
till he Thred fall on 100 parts in the Quadrat of 45 gr. in the Quadrant,

|
|
fo flaall the Height of the Place above the Level of the Eye, be equal to

‘
Ii

If che Thred fall on §@ parts of a Right Shadow, the Heightis but half
itis three quarters of che Diftance, For as oftas the Thred falleth on the

“parts of Right Shadow,

itt

es} 60, to the Parts on which the Thred falleth: |

S0,is the Diftance, to the Height required, ies |
And om the contrary, 2 |
As the Parts cut by the Thred are to 200 = 3 ape

So the Height, unto the Diftance, na
But

ke HS He, gee Vee a ee HG) Te aS GUL TN rs

is But when the Thred thall fall on the parts of Contrary Shadow, if it
fallon go parts, the Height is double unto the Diftance 5 -ifon 2§, it is
four times as much as the Diftance. For as oft as the Thred falleth om the

‘parts of Contrary Shadow
‘parts of Contrary Shadow,

| Ass the Parts cut by the Thred, are unto 100%
| Sois the Diftances untothe Height.

‘And on che contrary,

As 100, are unto the Parts cut by the Thred:
Sois the Height, anto the Diftance.

junderftood of the Height and Shadow. :

3: To find a Height or a Diftance at two Obfervations.

Asif the Place which is to be meafured might not otherwife be ap.
‘proached, and yet it were required to find the Height BC and the Di-.
flance ¢ Firft, I makechoice of a Stationat A, where the Thred may fall”
‘on 100 parts in the Quadrat, and 45 gr. inthe Quadrant, the Diftance

AB will be equal to the Height BC: then if I go farther in a diredt Line

with the former Diftance, and make choice of a fecond Station at D,where

the Thred may fall on 50 parts of Right Shadow , the Diftance B D™
Mhould be double to the Height BC; wherefore I’may meafure the dif-

ference between the two Stations A and D, and this difference D A will

beequal both to the Diftance A B, and the Height BC.

Or if I cannot make choice of fuch Stations, Ttake fuch asI may, one
‘fat D, where the Thred falleth at 50 partsof Right Shadow 3, the fecond at
E, where it falleth on 40 parts; and fuppofing the Height B C to.be 100),
Pfind chat, :

As 50 Parts are unto 100, the Side of the Q#adrat :
$0100 the [uppoled Height, unto 200 the Diftance DB.
And as 40 Parts at the fecond Station, anto too.
$0100 the (uppofed Height, unto 250 the Diftance BE,

| "Wherefore the difference between the Stations D and E fhould feem
tobe 503 and then if in the meafuring of it I fhould find it to be either
more or lefs, the Proportion will hold, As fromthe fuppofed Difference,
7 t®

‘The veof the Quadvati = tye

i And what is here {aid of the Height and Diftance, the fame may be.

Re oa.> OF Ar,

3128 ThY fi fe the BR
fo che meafured difference 5 s fofrom Height to Height, and from Diftance
to Diftarice.
As if the difference becween the ewo Stations D and E being meatus.
red, were found to be 3e, ,

As 50 the [uppofed difference, unto 30 the trae diferent :

{ Seucothe (uppofed Height, unto 60 the true Height :
And 200 the (uppofed Diftance, unto 120 the trae Diffance :
And 250 at the fecond Station, nto 150 the Diftance BE.

The like reafon holdeth in all other Examples of thiskind. And if an
Index with Sights were ficted to turn upomthe Cencer, ic might then ferve
by the fame reafon for the finding of all other Diftances,

-|

A SECOND.

‘APPENDIX

| CONCERNING THE.
Description and UsEs

OF ANOTHER

QUADRANT,
_ ~ Fitted for Daily Practice;

For finding the
HOUR and AZIMUTH,

AND

Dher things of the Suns Courfe, in reference to the
| Eborizon, with New Lines, ferving to the fores
_ mentioned, and other od ne more accurately.

; avented a Mtr: SAM, FOSTER, fometime Profeffor
of Aflronomy in Grefbam-College.

The Defeription of the OU ADRANT.

TN Oncerning the Making and Ufe of this Quadrant, you are to
-___underftand, Thar the Hour and Azimuth-lines are like chofe
chat are fech upon the former Quadrant, and the Lifes are (ncft
irc) Es fame ; and therefore we lightly pafs them over, as is feen in the

Rrr fecond

129

yo Of the Quddrant,

iaeNIA

| «BEAN
| The Forefide CZ L fi, )
QUADRANT. Vi :

were Sf

_ The Backfide
OF the.
rT nn ere

Laie Tia | Re ae ad : Si od ;
the Ufes of the Ouadramts = tg
fecond Propofition: Buc the diftance between the Equinodtial and che
‘Tropicks is here fhortned, tothe end chat more room might be gained
aboves for the better placing, and the more exact dividing of the Equi-
“nottialsy which in {mall Inftruments may be divided to éach fecond De-
gree; and in larger, ro each fingle Degree,

If ic be required to make chefe yet larger, then may the fore-
mentioned Azimutli-lines be left quite out : for the ufe of them, as
they are here defcribed, is of {mall moment, very hardly making
good the Suns Coaft to one entire Degree; and: for ferious Prattice,
the new Lines added are far moreé fufficient. If this be granted, then
may the Equinoétial ftand below, by which means they (hall become
large enough, even in {mall Inftruments. Efpecially this may moft
fairly be done, if the Hour-lines be reverted, by changing the Places
of the Equinoétial and Tropicks 5 chat is, ‘if che Equinodial Alci-
tudes be inferted below, on the Circle neareft che Limb, and the
Tropical Altitudes above, in che Circle neareft to the Center. Thus
becoming more large, they will fupply all intended purpofes very
well. ; |

,

| There is no Scheme given of this change now mentioned, nor of
‘the Vulgar Hours and Azimuths, becaufe thofe Lines are well enough
known already , and this mutation is eafie to be underftood : Bue
}

i

for the infcribing of che new additional Lines, cake chefe following
Tables and Directions.

_ How to infcribe the Additional thes upon the
! QUADRANT.

I. For the Lises on the Forefide.

; we two Equinoctials DC and EB. the one for the Hours, the

i other for che Azimuths, are to be divided from the equal Limb
by help of the following Table; and are beft to be numbred from
the clofeft parts of chem to the wideft, as is done in the Figure.

|

Rrr2 ATable

132 To inferibe the Addstional Lines on the Quadrant.

: Wie das A 2 Mg

el Table of Eguinottial Altitudes, both for Howrs and) |
Aximuths, which ave to divide the Equinottial. |

| ‘Degrees spon the Degrees spon the; |
Equal Limb for | | Equal Limb for 4
the the et
Hours. | Agimuths| ~ Hours. |Azimuths :|
Dege.i|D. M.|/D. Mo. | Dee.|D, MID. i
> ie Sa © ie Se 1 48 127 33/30 35
442 29| 3 II SO.u2oW Soler nor
6|3 4414 45 52/29 23/32 4
8|4 58) 6 I9 $4 [30 14/32 46)
TO duh, L2hi7 0 $2 SO (31 4/33 24
VES OAD i281; Mae SOc 3L). Selah ee
14| 8 39]10 54 S60. [32 37/34 34
16.) 9. $3112 22 S$ 62. 133 21135 5.
18 |1T 6/13, 49) VR 64 [34 «1135 34)
290 112 18|15 «13 W 66 134 40/36 oo
22 [13 29/16 36] |8 68 35 15136 25
S 24 |14 40/17 56) |s 70 [35 48/36 47
26 115 SOTO 131 1S 72 136 18137 6
28 17 0120 29] 18 74 136 45137. 24
30 18 8j21 41 & 76 (37 10/37 go}. |
32 ION, LOIZ2 FT Q 78 137 31/37 §3
34. |20 22/23 §9 80 137 49/38 4
36° g2T 28125 - 4 82 13 3138. 14) el
38 [22 3226 5| 84. ie -1§/38 21
40 {23 39127 Sy 86 |38 23/38. 26
42 124 37/28 21: {| 88 138 28938) 29 };
44425 37128. §§ 90 |38 30/38 30
46 [20 36|29 47 | - |

The two Equinoétials being divided,
15 Make EN parallel to AC, andO D parallelto AB, .

ae

To inferibe the Additional Lines on the Quadranz, 12
~ 2. Make ENa Tangent of 45 deg. or Radius ; then fhall AM be the
Co-tangent of the Latitude, viz. in our Example 38 deg. 30 m.
3. Make N A Radius ;then thall A P beche Tangent of 23 deg. 30 m.
and the Line A P to be fo divided into 23 2 Parts. - _ Af OREN
“ 4. Making AM equal to the Co-fine ef the Latitude, A © hall ‘be
_ the Sine of the Latitude. |
5- Make A X equalco A M on both fides.
6. Make a perpendicularto A 2 603; and A @ equal to AN.
7. Aris the half Tangent of 7¢ deg. S
8. M A being made the Co-fecant of rhe Latitude, find the Radises
_ thereunto belonging, which Radiwe make a Tangent of 45 deg. then are
_~ the Hour-points upon the Side A C the refpective Tangents of 15, 30,
45,and 60 deg. :

9. Draw rs from the middle Point of M-A, and draw the fifth
Hour from M parallel tors: And alt the reft of the Hour-lines muft-
| be drawn from their feveral Points to M as their Center, The Line of
6 isdrawnfrom the Center M, perpendicular to A B; or parallelto AC:
_ And all che other Hours beyond 6 may be tranfferred by a Bevel Square.
| 10. The double Square of 23 4 equal gr. isdone thus: -

Add 20 gr. to the Equinoétial. Alurude ; infere che Sum, and the
Equinoétial Alcitude :- Divide che intercepred Arks into 20 equal parts,
to which add 3 and a half of the fame parts. This is-to be fet both
ways from the Equinoctial, upward and downward, which the inferted _
Tables willhelp you to do. ;

Il. Fir the Lines on the Back fide...

r, On the Back- fide, ler the Points M and © change Places (or be fet
contrary to what they areonthe Fore-fide) and then all che ocher Work _
(forthe manner of it) is the fameas on the Forefide.

‘ 2. Forthe reverced Hours, cake every Hour- point (upon AC) from:
— Ayand turn it twice upon the 6 a clock Line from M, through which Points:
(and their Correfpondents on the Line A C) draw the reverted Hours.

3. The Scales tor the Suns Declination; and Months, are inferced fromy
fuch Tables as are common. |

4. The Limb for the Slope-hours may be about a feventh‘or eighth part?
of the Radius; andthe Marginal Divifions numbred 1, 2, 3, 4, 5,6, &e.‘
for Scars, muft be put imby that Scale of Declinations according to which \
you pac inthe Hours and Azimuths: And the Stars may be fuch asin the
a | following,

7. 44%. See 2 eh a eH ed ‘> no re ns ee a "EYE hea » ha ; r
iy GOR), aad “ett aer) Pe hey a ; AME eo aN Ut SUMO Cine ale Re NN vee he a a
iad f i NWP Coe oi u 4 ela ae pn es eta, 2
x ° hoe PR SAN ; ;

‘134 — Toinferibe the Additional Lines onthe Quadrant, |
following Table, or fach other as any thall defign to ufe ; but thofe were

conceived by the Author to be as felect as any, they being (one or more of _

them) always in view, and fic for obfervation,

N _ Names. R.Afc. (Declin, | M | °
§ -|Exiva ale Peg. “| O 26/13, 27) 2
6 iCauda Leonis 3 281/16 Pat Oy see /
5 Cor Leonis 16 BIl3. a7? I
| 3 {Os Pegaft | Ip) S588 ral 3
5 apa eae a
I | Procyon 34. 431 O80 SY oe
ee | Dex. Hum. Orion 47 401 7° 2402
4 \Cap. Ophix. sO 212 52| 3
2 | Med. Nex. Coll. Se-p. (63 58} 7 24.072
9 |LucidaPleiad, — 6 0123 Fhe F
8 | Aréturus ‘ 74 $3/20 58} 1 |
7 woth Y prec. 78°. 3447 | 40.
pg cp ee

was communicated by Edward Page, living at the Signof the |
Sugar- loaf in Hofier-lane, who maketh this and all other Ada-

om i Di Conjtruttion of this Ouadrant, as it is thas metamsorphofed,

thematical Inferuments.

TF otherQuadrants were thought complete in ufesthis willbe found much

more copious: For it ferverh not onely to find the Hour of the Day

by the Sun, of the Night by the Stars, and what elfe belongs to their Rie
fings, Settings, Amplitudes, @c. but is very well fitted alfo co defcribe all
the moft ufual forts of ftanding Dials; that is, all that are upright, or -

elfe reclining or inclining to the full Eaftand Weft; which two forts will
_-furnifh many kinds .of fuch Bodies as are regularly formed. Thefe are

here performed by very eafie and familiar ways of working, The No-
_ @urnal for the Hour by the Stars, is more expedient inthis than in other
Quadrants : For in judging of Time onely by the Appulfe of the Stars
to the Meridian, and finding that Meridian too onely by a rude conje-
éture from the Norch-Star, an errour of a quarter or half an Hour is eafily.
unawares committed, This cannot be fo here, if any ordinary care be’

had

‘ward; and Fanwary 30. it declines about. 14 gre 30 my Southward.

i The Ujes of the Quadrant, —— 129°
Kad in taking the Stars Altitude. For this purpcfe there are twelve {e-
le&t Stars inferted, allof them of North Declination, lying between the
Pauinostial and the Tropick of Cancer ; and in (uch difference of Right
Alcenfions, as that one or other of them will be always in fuch convent-
ent place of che Heavens, as from whence the Hour may very fully be col-
lected every Night throughout the whole Year. Since therefore they are
fo convenient for ufe, there would bea lirtle more diligence ufed to come —
to the knowledge of them in the Heavens, that due Obfervations may be
made whenfoever any of them fhallbein view. If any defire thac other
Stars (fuch as-are better known to them) fhould be inferted, they may.
have their defire eafily fulfilled : Onely chey muft cake care, thae the Stars _
be fuch as fall becween the Tropicks in the Heavens, and chiefly between
the Equinoétial and North Tropick, becaufe {uch Stars are longeft in view,
and their Hours beft found. The Propofitions that are here fee down,
might have been encreafed both in number and in variety of performance,
if perplexity had been affected ; but fuch of them, and fuch ways of ef-.
fecting them, arehere picched upon, as feemed moft conducible for daily

ufe. And for the fame reafon itis, that che feveral Lines upon the Qua-
* drant are denoted by Letters onely, that by fuch brevicy all unnecefiary
~ circumlocution might be taken oft, which, by impofition of Names to -

each of them, could not fo eafily have been avoided, ,
Tf the former Quadrant have heretofore found good acceptance,becaule.

- itisof fome good ufe, Idoube not but a greater proportion of chanks-
willbe given from the Ingenious, for making publick this larger Improve-_

menc of this Inftrument.

The USES of the QUADRANT.
I. To find the Suns Declination.

> Ay the Thred tothe Day of the Month upon the back- fide of the |
L Quadrant, and ic will thew youthe Declination of the Sun in thac |
unequal Scale, which is mumbred with twice 3. If your Day fall in the
upper Scale of Months; (which may be called the Summer-fcale) then is;
the Declination North : If ic fallin the lower (or Winter) Scale, the De-
clination is South from the Equinoctal. |

Thus upon 4pril 20. you thall find the Sun to decline 15 g7. North ,

a

ie i

136 ‘ the Ufes of the Quadrant. | }
q Thecontrary Work is eafie; by affigning the Suns Declination, to
know on what Day of the Month the fame thall be. For the Thred may P
be laid to the Declination in cwo Places » In both which ic will crofs the
‘two half years, fhewing two feveral days on which the Sun fhall have fo _
‘much Declination North: and two days more; on which it (hall have
that Declination Southward. It will be eafie to diftinguifh which of

thefe days ferves your purpofe, by the two Seafons of the Year, unto —
which the two Scales of Months do an{wer.

IJ. Yo rethifie the Bead for Obfervation of Hour or Azimuth: and
to performs thofe things that are done by the ufual Lines upon the

Quadrant,

Hiv found the Suns Declination for your Day, you muft count
A A: the fame upon the double equal Scale which is on the fore-fide of _
the Quadrant, namely, from the middle of it cowards che righthand, if
the Declination be North, or towards the left hand if icbe South, The —
Thred being laid thereto, you muft move the Bead till it fall juftly upon
the Hour of 12, fo thall ic be fec right for the intended utes of thacday .
As, *
1. For the Hoar. If you obferve che Suns Altitude (by lettingtheSun- |
beams to fhine through the Sights, and the Plammet tc hang at full li-
berty clofe to the Plane of your Quadrant) che Bead will thew the Hour, ~
if you have refpec to che time of the Year: That is, If che Suns Decli- ~
nation be North, the Bead fhewsthe time of the day among the Summer- —
hours, thofe which fpread from che Equinoétial tcowardsthe right hand.
If the Sun decline South, the time mut be accounted in the crofs Lines, —
which are the Winter-hours. And in this Obfervation you fhall feethe
Thred to cut (in the equal Limb) the Suns Altitade above the Horizon.
Thus at London, if the © decline 15 gr. Norchward, and the Alti-
tude were 9 4, gr. the Hour would be about a quarter before 6 in the
Morning, or a quarter paft 6 in che Evening. Bucif the Sun had che
{ame Declination Souchward, and the {ame Altitude alfo, then would the
time be half an Hour paft 8 inthe morning, or half an Hour paft 3 in the
evening. The former of thele times is fhewed by the Bead among the.
Summer: hours, the latteramong che Winter-hours. |
(26 For the Azimuth. If the Suns Alticude be numbred the contrary
way in the equal Limb, and the Thred be laid thereto, the Bead will chen
Shew che Azimuth of the Sun, if you account it according to rhe time of

the - |

< ‘To vettifie the Bead, and find the Amplitude. 137
the Years that is, among the Summer-azimuths when the Sun hath
- North-declination, and among the Winter-azimuths when the San de-
_ clines South. The Summer-azimuths are thofe thar. {pread from the
» Equinoétial cowards the lefthand; the other croffing them are the Win-
_-ter-azimuths. Thus if the Suns Declination were 8 gre Northward,
_ and the Altitude 18 gr. the Azimuth would be 80 gr. trom the South :
_ Bur if the Sun had 8 gr. of South Declination, and 18 gre Altitude. the
- Azimuth would be sogr. fromthe South here at Loxdon. This way
_ may ferve for grofs works, when the Azimuth is required onely wichin
one or two whole Degrees. You fhall find it done more accurately, and
for better purpofes, in the thirteenth following.
32 For the Afcenfional Difference. The Bead being rectified as before,
and applied to the left fide of the Quadrant, gives the Afcenfional Dif.
ference, or the time of Sun-rifing and fecting, before or after 6 a clock,
among thofe Hours and Quarters which interfe& each other upon the
fame lefe fide of the Quadrant, if you: count chem agreeable to the time
of the Year: And from the Bead to the Line of 12, rightly caken, ac-
cording ¢o your time of Summer and Winter, gives the Semidiurnal
_ Ark of the Sun, or half che Days length ;:——=-As alfo, from the Bead
tothe other Line of 12, which ferves for the contrary time of che Year,
‘Bives the Semineéturnal Ark, or half the length of the Night.——Thus
_ if the Suns Declination were 14 4 gr. the Afcenfional Difference would
bet Hour and 4 of anHour: And if thefaid Declination were North,
then the Sun rifech thae day * of an Hour before 5, fecteth 4 after 7,
The Semidiurnal Ark (fromthe Bead to the Summer 1 2) is 7 } Hours.
_ The Semino@turnal Ark -(from the Bead to the Winter 1 2) is 44 Hours.
Thefe doubled make the day 14 3 Hours long 5 thenight 9 3 long.
4+ For the Amplitude, The Bead applied to che right fide of the Quas
drant, gives the Amplitude of Sun-rifing and fetting in all varieties:
Namely ; From the Bead to that South-azimuth which is proper to the
Seafon of the Year, is the Amplitude from South ; asalfo, to the contrary
Souch-azimuth, givesthe Amplitude from North: thewing how many
Degrees of the Horizon the Sun rifeth and fecceth any day from the juft
South or North, So from the Bead to the Eaft and Weft-azimuth (which
is the ninetieth Azimuth) gives the Amplitude from Eaft or Welt.
Thusif the © decline 14 2 gr. the Amplitude ishere 23 4 gr.almoft. If
the Declination be North, chen is this Amplitude from Eaft and Weft
towards the North 23 4 Degrees. The Amplitude from che North ic
{elf isthen 664 er. From the South point of ye Horizon ic is 113 re
Pay. S{ You

238 To find the Legisning.and.endof Twilight) =
You may eafily (in fuch manner) account it for South Declinations of |
the ©. rage

~~ at

V. To find when Twilight begins in the Morntng, and ends at Evening a
which Moments are the twoutmoft Terms of Dark Night. =

A. Frer the Bead isreétified for your Day, the Thred laidto 18 gr. in
& the equal Limb, will thew che Hour or pare required. Only here |
remember to take your Hour aright: Namcly, in Winter cime look |
among the Summer-hours, where ic is that che Bead sreftech 5 for that is”
the Morning or Evening Hour of Twilighc: So in Summer time you mult
look among the Winter-hours Thus when ctheSun declines 12 gore
Southward, the Twilight begins at London ac § in the morning, and ends”
at 7 aclock at night, asthe Bead fhews among the Summer-hours: But
if that Declination were North, the Twilight would begin ac 4 of an
Hour before 3 in the merning, and end ac { after g at night.————The |
Suns depreffion 18 gr. under. che Horizon, isthe ufual Term whereon to
begin and end the Twilight. You may as well do thisto any Degreeof
Light, asto 42 or 13 Degrees depreffion s at which time in the morning |
all chings begin to be vifible, and the Light to be ef fomeufe. As if the:
Sun decline 3 4.gr.Southward, if you fet the Bead thereto, and then lay
the Thred at 12-gr. in the equal Limb, you thall fee the Bead (among }
the Summer-hours) fall upon 5.in the morning, and 7 at night ; fo chat |
at §, andcill 7, there isa reafonable degree of Light. Or if in Summer
the © had declined 7 4 gr. Northward, the {aid degree of Light would |
begin at 4in the morning, and end at 8 in che.evening. Near to

ay

the longeft days you (hall find no Twilight at all, according to 18 De>
ercesdepreffion of ©.underthe Horizon; for then the Bead will fall be- |
yond the Winter 12 a clock Line, odo aid

@ Thefe arethe chief Ufesof the Hour and Azimuth-lines, as they
are here, and in all Quadrants commonly inferred, There are other
things, concerning the Suns place in the Ecliptick, the Suns Declination, |
the Suns Right Afcenfion: Namely,—How by having any one of theft,
to find ouc the reft.—Thefe are here omitced, as matters onely of curio=
ficy, being of no furcher ufe in this Inftrument, than that chey may be |
known: Yer if any fhould defire them, chey may havea Scale of the 12.
Signs infcribed on the back fide, by help of which, the fore-named requir
fites may be actained. ,
_ The Parciculars chat follow are moft aimed at, (as being more of

chem,

ie? . AR, 1 a Meal cesta ‘ eae |
ie To fiad the Afcenfional Difference; and Amplitude. %39
them, and’ more’ accurate) and therefore the precedent things are clus
| briefly paffed over. | |

TL. To find the Suns Afcenfional Difference, Bec,

AR

Behe the Declination in the equal, Limb from Fro K: The Thred
“Nu there laid gives BS the Afcenfional difference :— The.faid AfcenGi-
onal difference gives the.cimes of Sun-rifing and fetting before and after 6,
whofe Declinations are known.
ee j

f
}

_ @ So by having the Afcenfional difference, you may find the Suns De-
‘dination thereunto belonging.

‘is 27 gr. 14», chat ist he. 49%. And if chis Declination be North, the
‘Sunrifech 4p. 49 m. before 6, and fetceth fo much after 6: that isp ic
rifech 11 #. after 4in the morning, and fectech 49 m. after 7 a clock at
might: And the timeof fetcing being doubled; gives 15 be. 38 w, for the
dayslength: The time of rifing being doubled, gives 8 ho. 22 m. for.the
Tengeh of the night. But if the Declination had, been South, che Sun

!

49 m. before 6,(that is,at q,and TI m.) and the day would be $ ho. 22m.
long ; the night, 25 be, 387. : 7

it
i

k

IV. To find the Suns eAmplitudes 8cc.

NOune che Declination in the equal Limb from Gto H3 The Thred,
‘pe Stars whofe Declinacions are known.

| @ So by having the Amplitude, you may find the Declination: For if

fl

the Amplitude be counted from C to R, the Thred laid arR gives che
DeclinationGH, |

At London, if chi Declination be 20 gr, the Amplitude is 32 prs 20m,

from the Eaft and Welt Points of ‘the Horizon,

I's aeteae ‘
| _V. Having the Declination of anyupright Plane, to fond the Elevation
| ' of the Style, 87c. eit

oe the Thred co the Planes Declination, counted from Dto R: fo
| . will G H be the Elevation. | )

|
|
)

owith che lengrhs of Day and Night.—The fame may be done for all Stars..

| Here at London, if the Declinntion be 20 gr. the Afcenfional difference

)

| |
fhould rife.t ho, 49 m. after 6, (that is, at 7, 49 a.) and fhould fer 1.40
|

¢ there laid gives CR for the Amplicudé.—The fame may be done |

S{f2 q So.

an
‘ ¥ cr . ote
> an

140... Propofitions im Dialling.
_@ So by having the Elevation G H, you may find DR the Declination,

Tf an upright Plane (here) decline 20 gr. che Styles Elevation will be ;

35 gr. 48 m, | :
‘VIL To find the Deflettion, oc.

) “Ount the Dahaanee from BtoS: The Thred there laid gives F K

the Deflexion. —

q So by having FK the Deflexion, you may find BS the Planes Dee

clination.

If a Plane declining 20 gr. the Deflexion is 15 gr. 13
VIL. To find the Difference of Longitude, Gc.

Ie Digi the Elevation from F to K: ES is che Difference of Eon.

gitude. ae
2. Count the Deflexion from Gto H: CR is the Difference of Lon
gitude. |

| may find the Elevation and Deflexion. git
A Plane declining 20 gr, hath 25 gr. difference of Longitude.

VIII. To wake an Horizontal Dial,

See 2 Ount the Hour from E toS; che Thred laid at § gives F K. Thea |

q By the contrary Works, havingthe Difference of Longitide, you |

ay. |i
ie |
ora! |i

count G Hf equal to FK; the Thred at K laid gives DR, the |

{pace of that Hour from t2.

2. Count the Hour from CtoR, and by help of the Thred you (hall |
haveGH, Then count F K equal toG H3; the Thred laid at R, gives

BS for the {pace of that Hour from 13.

3. With a pair of Compaffes take the Hour from C to R,and fet it from, |

BroS: BSis the Space or Angle of that Hour from. 12.

4. Take with your Compaffes che Hour from E to S, and {etitfrom D
toR: Sothenumber DR thews how many Degrees that Hour muft be _

trom 1 2.

By all thefe ways (here at London) the third Hour will be found about

38 gr-fromt2, Thereft will be in like manner found according to their |

true quantities.

S

IX. Te

sae |

|

‘lee Laue
ili > eee
|
|
i|

Propofitions in Dialling, «RAE

IX, To find what Angle any Hour-circle maketh with the Horizons or
| any Azimuth makes with the Equinottial.

| Ec the number of the Hour-circle (or Azimuth) from South, be
Ly counted fromC to R ; the Thred laid at R willcut the equal Limi
‘in H, and F H will be the Angle required.

.. q By che Angle knowa, it will be eafie, by the contrary Work, to find
the Hour (or AZimuth) to which that Angle belongeth.

© The third Hour (or 45 Azimuth) makes with theHorizon (or with the
_Eguinottial) an Angle of 36 gr. 55 ms. here at London. ,

X. To find what Ark of any Hour-circle is intercepted between the Equi
noltial (or any Parallel) andthe Horizon.

| gNOunr the number of the Hour-circle from South, from Eto, or, if
it be above 90,from E to B,and back again to S : So FK in the equal
Limb willbe che Ark required, between the Equinoctial and Horizon.
-. The Ark intercepted between any Parallel and che Horizon, may hence
alfo be found.—If the Declination of the Parallel be North, and the Hour-
be between 12 and 6, add the Declination tothe Ark found by che former
“Work: In other Hours beyond 6 fubtra& the former Ark out of the De-
dination, the refule will be the Ark required. Upon the Hour of 6 it felf,
the Declination of the Parallels is the Ark intercepted.—If the Declina~
tion be South, fubrract it out of the Ark found before, (namely, the Ark
intercepted between the Equinodtial and Horizon)what remainsis the Ark
intercepted becween that Parallel and the Horizon.
| Thus at London, the Ark of the third Hour intercepted between the
Eguinoétial and Horizon is 39 gr. 21 m.—And if the Declination be
18 gr. North,the Ark intercepred between that Parallel and che Horizon is
47 gr. 21 m—If the Parallel be 18 gr, Souch,the Ark will be 1 8 gr.2 tm.
- @ The firft Werk will alfo thew what Ark of any Azimuth from
South is intercepted berween the Horizon and Equinodtial, if in ftead of
‘the Hour-circle from South, you ufe the Azimuth from Souths. This
intercepted Ark is the Equinoctial Altitude of that Azimuth..
Sointhe 45 Azimuth from South, the EquinoGtial is 29 gr.2t a. high.
nthe 235 Azimuth from: South, the Equinogtials depre(fion under che
Horizon is.29 gr. 21 7%. ;
This is made ule of afterwards,

g rer: - XI. Hom:

142, To find the Heightof the Sun spom-any Azinsuth.

HAtbOMN

XI. How high the Sun. fhall be upon any Aximsth, and’ in any Decli-

T He Azinauth is beft numbred from the South : And this Propoficion |

(with moft of chofe char follow’) is done by help of Compaffes, —

_ But if the Sun have Declination, then firft lay the Thred from E to-

wards K, according to that Declination, and take the leaft diftance from

the Point B to your Thred, and keep this.extent. Then,
@ It the Suns Declination be South, count your Azimuth from E to

‘q If che Suns Declination be North, and

and leciecut E Nin T ; Then fet one Foot o

30m.North, his Altitude upon the 45. Azimuth will be 423 gr—But

if the Azimuth be more than 92, count from BtoS, the excels above 903
and applying che Thred thereto, fee what

Thred cuts from F. Count that number. of

{uppofe tocut the Line # in we

rft extent)upon. «and turn the other Feot towards the Side A C; laying.
the Thred ac the remoreft turn,

required, Thus.if che Sun declj
be 8o4 gr. 15 m, from the Sour
our Laticude of 51 £7. 30m,

Degrees _
165 gre
the Azimuth lefsthan go.
from South, count your Azimuth from EtoS, and lay the Thred ac it, |

f your former Extene in Th,
and with the other Foot turned abour, lay the Thred at cheremoteft di.

AC: The Thred {o lying, fhews from B
in the equal Limb the Altitude required. Thus if the Sun decline 11 fie

Degrees of che equal Limbche

Degrees from 60 (inthe equal
Limb) forwards, towards 7°; 80, go, and lay the Thred there, which _

Set your Compafles (keeping ftill cheir

If now, to the Thred {o laid, you num-
ber the Degrees in the equal Limb from 60, the fame (hall be the Altinade

ne I! gr. 30 w#. North, and che Azimuth
h, che Altitude muft bess LP 45 rosin

eAnother

i! z eeceree £97 7 ey st te
lei

Be fod the Height of he Sunt all Hows, 14x
| Another way for this Propeftion.

¥ the firft Work inthis rich, geethe Equinoétial Alcicude or De-
| preffion for your Azimuth: Then lay the Thred ‘at E, andin C-D,.
from D, counc the faid Alcieude or Depreffions from which Number, or
Point, take che leaft diftance to the Side A C. Enter this length between.
the Side A C and the Thred, keeping one Foot upon the Line AC, and

“Yemoving it thereon coo and fro, cll the other Foor turned about may
| jaftly touch che Thred: Then keeping your Compafles there fet, re-
| move che Thred from G toward H, according'to the Suns Declination,
and cake the leaft diftance from your former ftanding co the Thred._ This:
length meafured in the Scale C D (fo as one Foot ftanding upon the Scale,

| the other turned about may juftly couch the Side A C) fhewsan Ark;.
| which ig

If che Suits Declination be South,

__ ‘muftbe fubtraéted from

‘FF the Suns Declination be North, othe Azimuths Equinoctial Alcieude...

and the Azimuth lefs chan 90,

~-muft be added'to |
Ff the Suns Declination be North, and che Azimuth more than 90, the:

_ Azimuths EquinoGial Depreffion muft betaken ouc of this Ark..

| The refulc isthe Altitude looked for,

\

the Equinoctial

j

Thusif the Azimuth be Je - from South,.
Altitude , aah
| Depreffion will be. 25 a. g7-
| The Ark found will be 14 4, Then,
Tf the @ decline 11 $South, the Alcicude upon the 70 Azimuth will
be x Degree. |
‘TF the @ decline 11 3 North; the Alutude upon the 70 Azimuth will
. be 29 3 Degrees. im |
If the Suns Declination were 26 gr. North, thac forementioned Ark
would be 25 gr. whence king 153, there remains 9.4 for the
Altitude of the Sun upoh.the 140 Azimuth from South, at chac
ii Declination of 20 gr. North.
_ @ By this Work may a Table of Altitudes be made, by which the:
former Azimuthelines upon the uadrant may be inferted. ant
| A | W, Fo

pen ESE eee oe
144 To find the Height and Azimuth of she Sun

XID. To find bow high the Sup {ball be at any Hour, and in any Des

¢lination,

find the intercepted Ark of your Hour Hour, between the Pas
~ vallel of Declination and the Horizon, by the tenth. R
Secondly, Find what Angle your Hour circle maketh with the Hori-
z0n, by the ninth,
Thirdly, Count that Angle from C towards D, and from thence take
the leaft diftance to the fide AC: Meafure this length upon the fide A Cc
(from A) and there fet your Compafles: Then keeping that ftation of |
your Compaffes, lay the Thred to the intercepted Ark, counted in the -
equal Limb from G, and take the leaft diftance from ‘your ftanding to the
Thred. | Set one Foot of this length in the ScaleC Dj fo as that the other _
being turned about may touch the fide AC 3 fo thall thar Foot in the Scale :
CD give the Degrees of Alricude required, if you number chem from C, |
Let the Hour be 3 from Noon: The intercepted Ark between the 4

Equino&tial and Horizon will be 29 gr, 22 m, And if the Sun de-
cline North Ir 3 gr, the intercepted Arks will be 4° 52 And the Angle |
South 1137+ the intercep aah dna ep gle

of the third Hour with the Horizon is 6 3 gr.53. So that the Altitude
for ala Declination of 144 gr. will be ae Degrees. i

q By this Work you may makea Table of the Suns Alcicudes upon |
any Parallel of Declination: And by thofe Altitudes you may infert chofe /
Summer and Winter-hours which are upon the Quadrant, |

XIIT. To find the Suns Azimuth.

Ft Lay the Thred to the Suns Declination, counted in the equal |
Limb from F coK, and cake the leaft diftance from the Point Bto
the Thred, and keep your Compaffes at that extent : Then count the —
Suns Altitade in che equal Limb from F, and lay the Thred to it. This
being done, | : |

q If the Sun decline South, keep one Foot of your Compafies always
upon the Line E N, beyond the Thred, towards E, and remove it fill
upon that Line, cill the other Foot being turned about may touch the ©
Thred precifely. Obferve then where the Foor of your Compaf{s ftandeth

upon the Line EN fuppofe at Vs Bring the Thred co V, and it fhews |
ty E) the Azimuth from the South, q If

ty oA. £4 AS ~~:
4 _ a - ¥

-

Le 10 find the Suns Azimuth, 4g
@ Ifthe Sun decline North, keep one Foot of your former extenr,
apo the Line EN, on this fide the Thred towards N, and remove it
fill upon that line, until che Foot chat is turned about do touch upon
he Thred. And obferve where your Compafs Foor then ftandeth, upon
heline EN ((uppofe it ftand at W ) Lay the Thred at W, and it will
~utthe Scale E B; the parts whereof, from E tothe Thred, are the Azi-
‘auth from South. |
~ But if it fo fallout in North Declinations, that when the Thred is
aidto the Alticude, you cannot find room upon theline EN, whereon
© fet your Compaftes fo as to keep the conditions before required 3 then
yorkinchismanner: Add always 30 degreesto the Suns Altitude, and
‘aythe Thred at that compound Altitude, numbredin che equal limb
tom F. To the Thred fo laid, enter che former extent of your Com-
afles between the Thred and the line # #, keeping one Foot always upon
jat line. ~And look where the Foot of your Compafles reftech upon
qat line, fuppole at *. Take then che length trom ™ to 4, and fer
- vpon the line N E (from N towardsE) and to the point where ic
‘fts, apply the Thred, obferving what parts ic cuts upon the Scale
om B. Thenumber of thofe parts givesthe quantity of the Azimuth
dove gofrom the South. Or the partscut from E, give the Azimuth
-omthe North.
@ If the Sun decline net atall, but isin the Equinoétial, then the
“Te Altitude from F toK (by help of the Thred thereto applied) gives
‘Sthe Azimuth from’ South, 4

—Tfthe Altitude of che Sun be 21 2 in the Equinodtial, the Azimuth
om South is 60 degrees. |

‘ftheSun decline South § yr. and the Alticude were 15 3 gre the Azi-
juth would be found 60¢r.
‘Ifthe Sun decline North 20 gr, and the Alticude were 50, the Azi-
tuth would be §0 gr.

Tf the Sun decline North 20 gr. and the Altitude were 92gr. the
wtitauth would be 310 gr. from the South.

@ If you fuppofe the Sun tohave no Altitude anddo work by tchefe
> © by thefe Rules, you hall find the Suns Amplitude, Ortive
and Occafive, from the South. As if the Sun decline 20 gr.
| North, you will find 123 gr. 20m, for the Amplitude from

the South,

taekss ek XIV. To

HPT PT Nan 7A ieee hs CRRA Sa RE ONS RL en
gies OP QU aa th) ns a

tA6 " To find the Hour of the Days i

XIV. To find the Hour of the Day by the Sine | was |

Ount the Suns Altitude in the equal limb frem F, and co the- Thred
Ci there laid, take the leaft Diftance from the poine By and keep this
D ftance. i : P ute ee

Then count the Suns Declination (which is had eafily by the firft
Propofition: ) from FP in the equal'lumb, and apply the Thred to ir.
Then further, 18 4 ae

@ If che Declination be South, fet ene Foot of your former extent,
upon the line EN (always on that fide the Thred on which E ftandeth
from ic) and remove it thereon, till the other ( turned about ) may
juftly couch the Thred AK. Suppofe (in fodoing) the Compafs Foot
ftayech ac V. The Thred applied to the point V, will cut the hour from |
Noon, ifyou count the intercepted parts upon E B, from E,—Thusif the

Sun decline 20 degrees South, andthe Alcicude were 13 gre $2 mm, the
hour at London would be ro or 2, on eo

@ Ifthe Declination be North, fet one Foot of your former extent
upon the fide AC, removing it thereon to and fro, till-che orher Foot
turned about, will only couch the Thred; When it is fo fitted,
that Foot upon the fide AC, keep its ftation, and from. thence ex
the other Foot tothe Suns Declination counted: inthe Scale A P. |
~ faft extent muft be applied to the line N E from N: and. whe
ftays, lay the Thred. Sothe parts cur upon the ‘Scale EB, will
the hour. But this maft be done with caution. For if chat Boo
that keptits ftation, ftood from A, beyond the Suns Declination in
the Scale A P, then the intercepted Ark from E to the Thred 5 give
the hour from Noon. Bur if the fore-named Foot ftood between 4
and the Declination, then the whole Ark EBgo, with the Ark fron
B back again tothe Thred ( thefe two put together) give the hour fron
3 Noon. |
Thus if the Sun decline 15 gx. Northward, and be 21 gryhigh, th
hour is 7 before or 5 afternoon. Or if che Altitude were 3.gre-the how
muft have been 5 inthemorning, or 7 in che evening: namely, 99 ant
1g degrees from Noon. : ‘oes ty © ‘|

|

| To find the Declinationof a Plane. ‘147
XV. On an upright declining Plane, to find the Angle between 12 and 6.
“grAOunt the Planes Declination from C towards D : From that point

‘to Y, wponthelineMY. The Thred laid at Y gives G K for the Angle
\between I 2 and 6. | } |

| Or count the Declination of the Plane from B towards E, and. lay
‘the Thred at-it. The Thred willcuc NE. Take from N to the tnter-
feGtion, and apply ittoM Y ; the Thred put co Y gives GK, as before.
If a Planedecline 20 gr. this Angle will be 66 4 at London.
fei XVI. To find the Declination of .a Plane.

ip

ck, draw an Horizontal line upon your Plane (which you may de
| by your Quadrant.) Then apply one fide of the Quadrant to that
line, fo as thelimb may be toward the Sun, and the Plame of the Qua-

‘Plummet,’ you muft hold that Thred clofe by the edge of che limb (Iet-
‘ting the Plummechang down at liberty) till the fhadow: of the Thred
afleth direétly through the Quadrants Center. Which done, you rhall
fee what degrees of the limb the fhadow curs from that fide of the Qua-
idrant which is perpendicular to the Horizontal line. This ts called the
Horizontal Diftance, At the fame moment of time, obferve the Suns
Altitude. . By this Alcicude you may get theSuns Azimuth from South,
by the thirteenth. ) 7

| > After chis Preparation, ‘take diligent notice; whether the fhadow of
the Thred fall betwixe che South, and the perpendicularifide of ,the

‘Souch and the {aid perpendicular fide (both of them) upon one coaft of
the fhadow. :

| In the firft cafe you muft add the Horizontal Diftance to che Azimuth.
nthe latter‘ cafe, you muft fubtract the leffer out of the greater. . The
refule (whether it be Sum or Difference) gives the Planes Declination
from the South. tt ee

|» Nore here in the fecond cafe. That if che Horizontal Diftance be greater
than the Azimuth, chen doth the Plane decline to that coaft (Eaft or Welt)
which is contrary to the coafton which the Sun ftood from the South.

‘This fallech out very frequently.
f 4

Tee 2 i, Note

7 take theleaft Diftancero the fide C A. Set thar length from M ~

‘idrant may lie Horizontally flat. Thirdly , having a loofe Thred and

Southend Or whether the fame fhadow fall fo, as to leave both the .

STL Ay ee
a ra Py
‘ ’

wy

148 To drave upright declining Dials: an
~ Nore alfo.in the firft cafe: That if che Sum of che Horizontal D Rance
and Azimuth do exceed 180 gr. then the Planes Declination from South
is contrary to thac eoaft whereon the Sun ftood. And itis found by
fubtra@ting the fore-mentioned Sum out of 360 degrees. This. hap-
pens more feldom; that is, only upon fome North Planes; and on
them, only then, when the Suns Azimuth is more than go from. the
Sonth ; andthe Horizontal Diftauce more chan is the Azimuth from
the North. .

Examples are here omitted for Brevicies fake. Only add this; thacif
the Planes Declination from South be above gagr. you mult fubdud it
out of 180, and che remainder is che Declination from che North. —By
this accounting from North and South, you may always make that your
Plane decline notabove 90: And as when ic declines nothing, itis a’
full South or North Plane ; fo if ic decline juft 90, ic is chen a full Eat
or Welt Plane. 7

ayy
ai

XVII. How to draw any upright declining Dial.

Irft, draw.a Perpendicular or Plumb-line A B,, and cro(s it at right
Angles with the Horizontal line BC, and make BA equal co AO.
in Your Quadrant. H |
2. Upon theequal limb of your Quadrant, count che Planes Decli=!
nation (from North to South) from G, and there keep che Thred : which |
will cutfome of chofe Lines that are drawn. within che upper Square. |
3. Obferve firft, thofe Incerfe€tions which the pricke. lings: anal
with the Thred at 4, d, a, Take then the Length from A, the
Cenrer of the Quadrant to 6, and {fee it here upon the Horizontal Line
from B tox, (always on that fide of B, which looks to the fame coal |
whereunto the Plane declineth.) So cake from the Quadrants Center |
A, to the fecond pricke Lines InterfeGtion with the Thred, acd; and»

i
|

fee it here from Bro2. So likewife the third Am, muft be fer from |
B to 3. | ay |
4. Obferve againall. {uch Interfections as are made with che Thred, .
by the reft of thofe lines whofe common Concurrence is in the.point M, |
namely, at 4, ¢,e,4: and take their fevera] lengths from the Quadrants
Cencer A, and prick chem here down on the other fide B (contrary to”
_ the coaft of Declination) namely, at rz, 10, 9,8.- Then for the ncxe
line upon the Quadrant (which doth.nor, but would interfe the Thre¢,.
if ic were drawn out far enough ) obferve. where the Thred cuts; the
| extras

i
7
i

i
|;
q |

be To drasy upright declining Dials. | 149
extravagant line rs, namely ins: and take from A tos, and turn thac
Length ewice from B, fo thall it defign the point 7. Afterwards at the
“point 7, draw the infinite line CD parallel to BA. Allo fet off the
hour of 6, on chat fide B which is contrary to’ the coaft of Declination;
namely, from B to E, according as the Angle between 12 and 6 hali

be found by the fifteenth.
Declination 28 deg. $. Eafte

eP | 5. Draw all the
\" : Hour-lines from A’, ee...
2 the Center of your
; | Dial, through the
ons 3, 2, ¥, 22, 0T,
ok ee 9,8, 7, in fuch
wife, that as many as
wellcan, may cutche
Ine DC, as is here
done, in pand gi ‘
6. Make6, 5, e- ;
qualto 6,7: and 6,4, ;
qual to 6, p, and 6,
3, equalto 6,93 and
|. craw che seft of the

: : ; N 7 S t
| / v8 6) 10H i 12, 2 2 3 Acurs 4 5,44, 43. ae
cs,

- Thus you may gec 12.

) hours, and if you ex-'
ls rend them beyond the Center, you fhall have the
a whole 24. Out of which you. may naake choice of

43 ) fuch as will ferve your ufe.

&

@) For placing the S syle.

4 Seek the Elevation and Defle&tion by the fifth and '
| fixth, And make BF equal to the Deflection; f{er-* ;
| ting che Subftylar line FA always on thar fide r2,'
which is contrary-to the coaft ef the Planes Declina-}
| tion. Make alfo F G equalto the Elevations So FAG will be the s
| paitern ofthe Style. 2

DP) PRO oe ieee OT Ree Rte) aE ig PE Ae ET Sana
wn

ago = «Of the upright full South-Dial, =
Or the Thred lying full at che Planes Declination upon the Qua-
drant as it did, Take the leaft Diftance from the point X to che —
Thred, and fet chat Length from B to H, and draw A H for che Sub-
‘ftylar. Then making AH Ka right Angle, take che leaft Diftance —
from: Mto the Thred, and make H K egual to this Diftance: SoisK AH
_ the pattern of your Style. 3
ae @ In-all Dials, The Style muft ftand juft over the Subftylar, eleva-
_. ted fo much above it, asthe Elevation (before foiind) cometh to. -
In South upright Decliners the Cenrer of the Dial is above (as in the
former figare) and the Style pointsdownward. Butin North Decliners,
the Center muft be low, and the Style muft point upward.

xe inden

XVIII. Of the upright full Sonth-Dial.

~w

“? fg te Angle between 12 and 618.90 degrets,.
ne The Line.of. £215 the Sub fteylarinien mW “srsnsssocdiieidneanies
The Styles Elevation is the Complemént of your. Latitude. ae |
The way of pricking down. the Hours is in'a° mariner, the fame with a

He Declination of the full South- Dial is nothing, “Whience it is, That

that before for Declinerse”” No moye needsto be faidvef it. ~~. . | aa
: The Ere&t full North Plane isthefame with: chis South. Onlythe
Style of this points upwards’ towatd the North Pole, as the former’
downwards towards the South Peleg "Fs 4 ok Sor baa
XIX:-Of apright far declining. Planes. naman n
i; ieee Dials are more difficule than thofe other Deciners ‘nencificedll
in the feventeenth, becaufe here the hours have no Cefter or Point of
-mMecting upon the Plane. Ic will not be amifs therefore to fet down the ©
‘i whole work in all parts of it. * |
i. Draw a Perpendicular or Plumb-line AB, and cro(s it ar right 2 |
_Angles wich che Horizontalline BC; Andmake BA equalta AQ in —
‘your Quadrant, fetting A above B if the Plane decline from the.Sor th, |
or below B if it decline from North | Tae |
_ 2 Count the Planes Declination from South or North, upon the
_ Limb of your Quadrant, from G ; and there keep the Thred. + 2
3+ Among thofe Lines on the Quadrant (whofe common Con-
currence is at M) obfervethat Interfection which is made by the fixth” |
Hour from the Quadrants Center with the Thred: Take the length

from

~—

from the fame Center co that Interfection, and prick it down here
from B to C (and om that fide B which looketh toward the Seuth,
4 the Plane decline from South; or toward the North, if che Plane
decline from North.) And draw out che Line C DE parallelro BA.

4 Obferve again upon that Quadrant the Interfection which the fe-
cond line from the Center makes with the Thred and take che length

from the Center of the Quadrant thereunto and prick it down towards -

€, namely from. B to F, e
An upright Plane declining
Eaftward.
§. Take the Lengths

from the Center of your
Quadrant to’every hour

C: and prick them all
down there, from C to

7 and §, from Ctro8and

Laftly, Take from the
- Cenrer of your Quadrant

that Length twice from
C: this double Length
will reach from C to 14,
at Ii. | hy
6. Lay a Ruler to A
and F, and transfer. the
point F unto H in the
| line CE. Then take the
Length from Hto 10, and fet it from A (towards
B) to 10, the fame way from A that 10 ftands
from H.

7» Wich the fame Length H 10 or Ato; go
to your Quadrant, and fetting one Foot of ir

Center, ( with the ether turned. about) lay

‘Of upright far declining Planes. — gn

82 degrees from South -

point upon the fide A>.

4, from C to 9, to, And -

to the point 7, and turn »

on the fide A C inthe fourth point from the.

the |

Soe
Pes 2-2,

eer P Ey MOM eee REO TOM, TO A ea
. De the Mm apt + ? Be ee oa
A oa a2 } j Ty .

(152 Of Forming and Placing the Stile.
the Thred at the remoreft D.ftance, and- keep it there. _ “
8. From every point on the fide A C of your Quadrant, take theleaft —
sd Ditances to the Thred folaids fetting chem down frem A to 7 and 3;
from Ato8 and 4, from A tog; A ro was put on before. Then the
: leaft Diftance from 7 to che Thred being twice turned from A towards B,
1 will give the Length from Ato rr, | .
: 9. For the finifhing then of the hours you have no more co do, but
. _ draw right lines through each couple of correfpondent points, namely,
a from 4 to 4, § to 53 fromCto A, or 6t063 from7 to 7y 808, 9
a to 9, 10tote@, and from a1 torr. | ,

I Concerning the forming and placing of the Stile. -

10, B* the precedent feventh Propofition you may find the Planes
Difference of Longitude, which (for this Plane chat de-
_ lines 82 gr.) will be (here ac London) 83 gr. 43 sin. and chac from
‘ the South, becaufe the Plane declines fromthe South, The Cemple-
~ ment of which Longitude (83 gr. 43 min.) is 6 gr. 17 min. Take
then firft, the Length from C to 7 the next hour point upon C E, and
_ carrying thatextent toyour Quadrant, fet one Foot of it upon 15 in
_ the Scale AP: and lay the Thred fo, that the other Foor turned a-
bout may juft couch or pafs over it, and keep the Thred there. Then
~ Gin the Scale A P:) count the fore-mentioned Complement , 6 gr.
\ 17 min. and taking che leaft Diftance from that Point to the Thred,
fec it from’6 a Clock at C, towards E if the Plane declinefrom South, ~
Cor towards D if the Plane decline from North ) as you fee it done,
here, atG. Secondly, do the fame work again upon the line A
Bs That is, take from A to 7 the neareft Hour poinc, and fet one
Foot of chat extent upon #§ in che Scale A P, and with che other
Foot turned about, lay the Thred as before. Then in che {ame
Scale A P, count the fame Number 6 £r+ 17 min, and taking the
leaft Diftance from thence to the Thred, fer that Length from A
to K, anfwering toC G. And laft of all, draw the Right Line
ee This fhall be the Line of DefleGtion over which the Stile muft
‘ftand, | .
tf, Furthermore, Through the: Points G and K (or any other
cwo points of the fame Line ) draw che two Lines G O, K P, both —
perpendicular to the Defleétion Line GK. Then confidering, char
every Hour comprehends 15 Deprees of Longitude (that is, thae
from

' Of Forming and Placing of the Style. = > 0533
“froma C t0°7 is 15, and from7 to 8is1§, ee.) and fince thar CG
is6gr. 17 min. IF CG be taken out of C7 which is ry gr. there
_willremainG 7, 8 gr. 43 min. Towhich, it youadd from 7 to 9,
| whichis two hours or 30 degrees, cheSum willbe 38 ¢r. 43 mn, whofe
Complementis 1 gr. 17 min. If now you make the Angles GMR,
-andK NS, each sa gr. 17 min. they willcurche Defle€tion Line GK,
in RandS. And if further, cto the Radius-G R’you defcribe the Ark —
-RT3 and to the Radius. KS you defcribe the Ark RT; and to the
‘Radius KS you defcribe che Ark S V5 and draw the Line T V, a:
-Tangentto both thefe Arks, the Trapexinns GTK V hall be the pac-
‘tern of your Stile. In placing which, you nuft be careful chae thele
perpendicular Lengths G T and K V (perpendicular I fay co T V the
| Faucial Edge) be juftly placed upon the two affamed points at G and
K.—Or having found G7to be8 gr. 43 min. you may add to it from
'Jto 10, which is (three hoursor) 45 degrees, The Sum will be 53 gr.
43 min. whofe Complementis 36 gr. 17 mix. If now from the points
“O and P (where the faid hour of 10 curs the two fore-mentioned Per-
« pendiculars GO and K P) you make the Angles GO Rand KP §, cach
equal to 36gr. 17 min. they will cut the Deflection Line G K in the
fame two points Rand$, After which you may proceed to make the
pattern of your Stile, as before. ,
| @ 1. Note, That in performing the fitth SeG&ion of this Pro-
pofition, inftead of taking thofe Hour points from the Cen-
| ter of your Quadrant upon AC the fide for your Quadrant
hag (if thofe Diftances fhould be too great of your Plane) you
| may lay the Thred any where upon the Quadrant, and in-
/ * ftead of taking from the Center to the fore-enamed Points, —
| you may take the leaft Diftances from the faid Points to
che Thred, feverally, and fet them down from C to 7 and
5, and from C to 8 and 4, and fo tog, 105 andfor 11,
you muft take from the Point to the Thred, and fec it
re twice from C3; by which means they will be all of lefs Di-
ye ftance from C. And then all the work is to be continued,
| as is before prefcribed. ——Or if the faid Diftances fhould
be too little, youmay donble, triple, or, ee. to make them
| greater. sane ey:
/ @ 2. Nore again, That in Decliners from the North, that Dit-
ference of Longitude which you find by the feventh, is to be
I-s -yeckoned from the North, and fo the Complement of it is
| Uuu 3 ®

i,
i

—

“134 sf Eafb and wefhupright Dial,

4 a i | VN ke oe ee Raw ey”, ie ee, ey Ld
E> Wh SB Ue ianite Seer ta 8.) SR ile aes Pah Oe te Tom
ee ik be i

m2
Vee

~ - tobe accounted from C (or 6 a Clock) cowards D. And
~ thae the wideft part of the hours in thefe North Planes matt.
poine upwards, and che clofeft parces downwards + contrary.

to what is exprefled here in this Plane, which hath itsDeclina-

tion from the Souch. 7 ones 5
q 3. Nore laftly, thar chis Direétion here given for enlarging the
_ Hoarsin far Deeliners, may eafily be applied to fuch Dire

or Horizontal Dials (as are mentioned in the 26. fyllowing).

upon which the Pole hath but {mall Elevation. For the Dial

(or only fome chief Hours of ic) being defcribed in its natural-
{tréightnefs, may be enlarged by the fame means that this leaf |

was. Which will not be hard to do, bue would be tedious

here to run over again, *

XX. Of full Eaft and weft upright Dials. :

“Wh Hefe are more eafie than the former fore were. For having drawn
the Plumb-line AB, and afflumed the Point A for the Hour of 63.
goto your Quadrant, and take from the Center of it to ‘all the Hours |
points upon the fide A C ; and prick the firft of chem down in the Line
A B, from A to 5 aid 7, che fecond from Ato gand 8, the third from)
A to 3, the fourth from A co2; and for the fifth, cake from the Center

_ of your Quadrant tothe Point r, and fet that Length twice from A, fo

it fhall Limit out the point 1. —Having thefe points, draw Lines through |
them, all parallel one to the other, and all pointing up tothe North |
namely, fo as to makethe acute Angles B A C equal to the Complement
of your Latitude. , é |

‘@ For the Stile. |

T muft always ftand overthe Line of-6 aClock, parallel toit, and
diftanc every where from it according to the: Length of AD. Which |
Length isfoon found, by drawing A D perpendicular to thé Hour- rine )
cutting the third hour from 6, inD. By which Line you may make the
pattern of your Stile. For the fiducial Edge lies parallel ro the Line of 6 i
AC, and at the Diftance of that Line A D,
1. Notchere too, chat if your Lengths from the Quadrancs Center
to the Hour-points be toe long, you may fhorten them by lay=

ing the Thred upon the Quadranr, according as your Conve-

nience |

EY EE

8
eF

Uuu 2

&&
“te
ae
te

&
<
$

£4o44404

” e Fuh ce 4 : ‘ ; . ; | e P : . | :
ie _ Of Eaft and weft upright Dialta == 55
| ~ nience fhall direét, and taking the leaft Diftances from thofe |
_ Hour-Points to the Thred; and fo pricking them on from A
( t0.6,,t0 5, 4, 3, Cc as was before mentioned tn the firft Note

/ upon the former Propofition. —Orif they be too little they may
-. be doubled, @&c, as is there exprefled.

$SSSSSSSESS
a

3 A full Weft
> upright Dials

q 2. Note further,

that what is here done

for defcribing thefe Eat —

and Weft Dials, may
be applied to the Di-
re& Polar Plane. On-
ly remember that you

are not tied (if the

Polar ) to make the
Hours co any fet An-

~ gle with the Line B

A, bue they are beft
at right Angles ; for
then the Line A B may

be taken for, and pla- .

ced as the Horizontal
Line of the faid Plane ;
all the Hours lying as
vertical Lines unto it.
And alfo the Line of
6 here muft be caken
(in the Direét Polar )
for the Line of 12, and
the reft of the Hours
are to be drawn alike
on both fides 12: no-
thing in {ubftance dif-

fering from thefe Eaft

and Welt Planes.

XXII. Jp

a i vw “nor. CS SS cS Ps Be eS. > PS = - 7 -
’ ‘ < : ? f a bu 4 Ge
a i ee Se vs
i. ‘ a aay Yi

16 Of Eaft andwef re-inclining Dials.

XX]. In Eaf? and Weft Re-incliners, to get the Defic tion.

Co the Re-inclination from D towards C. Take the leaft |
Diftance from thence to the fide A C. Set thae Length from |
- M to Y, and lay che Thred at Y. The Degrees FK will give the

Deflection. ‘
The Subftylar Line muft afcend in Recliners and defcend in Incliners,

from the Line of 12, accordingto the Quantiry of chis Deflection.

_ The Line of 12 lies always parallel to the Horizon.

XXII. To find the Angle between 12 and 6.

\ oan cei 1)
F aa|
& " x

“Ounte the Re-inclination from E towards B, the Thred there laid

will cut the equal Limb. The Degrees whereof from G to the -

Thred, are che Angle required
| XXIII. To get the Stiles Elevation,

| heed the Thred to the Re-inelination numbred in the equal Limb.
from F, and take the leaft Diftance from N to the Thred. Ser

one Foot of that length in B, and lay the Thred fo asto touch the other

Foot when ic is turned about. The Thred“fo laid, gives the Elevation.

in the equal Limb, from P.

XXIV. To find the Difference of Longitude.

1. ao the Deflection in the equal Limb from F, and lay
the Thred co it; and take the leaft Diftance from B to

the Thred. Put one Foot of this length in N, and apply the Thred |
to the remoreft Diftance of the other Foot. The Thred will then |
thew in the equal Limb, the Difference of Longitude, if you count |

trom F.

z.. Count the Defle&ion in che equal Limb. from G: and to the
Thred there laid, take the leaft-Diftancefrom B. Meafurethat length |
upon the fide ABfromA; keeping one Foor there fixed. Then lay |
the Thred cothe Planes Re- inclination counted alfo from Fin the e ual
Limb, and take the leaft Diftance from your ftanding to the Thred. —
Sec one Foot of this length in B, applying the Thred to the other.

Foot

—$——

/ How todraw the Dial, ‘157
_ Foot turned about. The Thred fo laid, gives the Difference of Longitude
inthe equal Limb, frem G. ; |
‘Thus ifan Eaft or Welt Plane re-incline, here at Londen, 30 degrees,
je will have in

Defles&tion—- ——— 47 deg. 26 m.
- Angle from 12 to 6 55 26
. Elevation-———> 22) 02

. Difference of Longitude—70 14

XXV. How to draw the Dial.
She the Back-fide of your Quadrant, in the upper pare of it,
ie

you-have Lines drawn altogether like thofe on the Fore-fide
placed near the Quadrants Cencer, the ufe of which was thewed bes
_ fore. ee
The manner of work in this Propofition is in. moft chings fuitable to
tharin the fevenceenth, and will need no other direétion.
Only for placing the Lines, Take notice, chat

| .

The line of 12 in thefe Eaft and Weft Re-incliners, lieth always para

el to the Horizontal line of che Plane. So that if we fuppofe the former .

“Figure of the feventeeuth to reprefent one of thefe Dials, then AB muft
be conceived to lie Horizonral, and BC Vertical. All other works will
“be like to thofein the fevenceench,

"The Stile in Recliners poineth upward, and the Subftilar and the
‘hourof 6 do afcend above the line of 12, fo much as the Deflection and

Angle from 12 to 6 come to. The Center of che Dial is on. the South end’

of the line of 12. 2
The Stilein Incliners pointeth dewnward, and the Subftilar and the

hour of 6 do defcend below the line of 12, fo muchasthe Deflection and
“Angle from 12 to 6come unto. The Center of he Dialison the North
“end of the 12. clock line. |
Thefe things being obferved , you muft count the Resinclination
BCs Plane in the equal Limb on the Back-fide from the lefe
Be sow ard the right , according as the Figures are fer: and there
Tay the Thred and keep ic. Then obferve how it cuts the Lines
‘next to the Center, and proceed in all chings as in the feventeenth

| before.

@ Note, That you may find the Inclination of a Plane by applying
one fide of your Quadrane to the Planes Vertical line: for fo che Didi
a , : wil

“558 To draw Eaft and weft ré-inclining Dials.

P 5%
will cut the quantity of Inclination in the degrees-of the equal lim
being numbred, from that fide of the Quadrant which couchech-the -
Plane. ——And for finding the Reclinacion, you may lay a Ruler to
the Vertical line of the reclining face, and cake the Inclination of the
under-fide of chat Ruler. That Inclination will be the fame with the
Reclination. es ‘A a

Nore allo, that this here delivered for Eaft and Welt Re-incli-
ners, is intended chiefly for drawing hours upon thofe kinds of Planes

when-you meet with them upon Bodies cut regularly. For others

wife you will hardly ever find any fuch juft Plane upon a fixed |

Building. |
Laftly, for a Scale of Chords, which here, and in fome of the prece-

‘dent Precepts isrequired, you may make ufe of the equal limb of your

~ Quadrant.

XXVI. To make an Horizontal Dial to any Latitude.

| ae draw the right Line BC, -and ereét che Perpendicular A Hy
# Then cake from the Center (on either fide of your Quadranc) |
to the third hour upon the fide AC; and:make AH equal there
co. And draw PH parallel toB C3 and the line 5K 7 alfo jolt in
the midft of them. After this lay che Thred to the Latitude of -
the placecounted in the equal limb: and take from every point of che |
fide A C, the leaft diftance to the Thred, and fet each of them down
both ways, namely, from A to4 and8, from A to 3and g, to2 and |
10, and from Atotandir, Then cake from the point r upon the ©
fide AC, to cheThred, and fet chat length from K to 5 and 7 both |
ways. —— You have now nothing more to do, but only from H to
draw the Hour-lines to allthe fore-named points: {othe draught is |
eafily finifhed, S |
The Stile muft ftand upon the Line of 12, and isto be clevated ace
ee to the Planes Latitude: as the manner is in all Horizontal —
Dials. : “4
~@ The ufe of this Propofition is to draw all Dials in any Lati-
tude for any dire re-inclining Plane, For, the Re-inclination com=
pared (in North Re-incliners) with the Poles Elevation: or (in
South direct Re-incliners}) with the Equinoétials Altitude, will eae
fily give the Planes Latirude: in the former che Difference was’ the
Elevation it felf: in-che later, the Complement of the Poles Ele-
i vation

ar

I a Z ie tu ry | T0 make an. Horizontal Dial. | “ 19"
vation. ———And this Propofition, with the fevenceench for uprights ie
| Planes; ‘the twentieth for upright Eaft and Welt, and fo alfo for 3
| Polar Planes on which the Pole hath no Elevation: the twenty fifth
Lia. ts « | a i ii

gy wt Anh io Pp

|
)
i | ; ee
)

| ie |

| for Eat and Weft Re-incliners: the eighteenth for full Norch and ~~ ,
South ere&, will furnith you’ with ways co draw ‘Dials upon fuch |
| ¥egular Bodies, whofe Planes have any fuch of the fore-mentioned
| Alpeas. , BY MMOs FO | ; |

XXVII. 1 find the Hour of the Night by the Stars.

|

ns
PIOHE Stars upon the Quadrant (one or other of them) will al- si
| “OR ways be in a convenient place of the Heavens: that is, of two or Me

’
ie

:

i %

i] * .

i 4

i

=e = ah. in quot ." ¢

160 To find the Hour of the Night by the Stars )
more hours Diftance from the Meridian. —— Having then made
choice of chat Scar thac is ficteft, look what number is annexed to the
name of ir. Seek that number in che left margin of the fore-fide of —
your Quadrant, clofe by the Hour-lines, and reétifiethe Bead to it.
Then hold up the Quadrant fteadily , with the fights levelled to -
the Scar, asif you were to take the Stars Altitude: and you fhall find

_ the Bead to thew (among the Summer hours of che Quadrant ) the

Motion of the Star in Hours, Quarters, and parts of a Quarter.
This is called the Stars Hour; bur this is not the Hour of the Night
ullic be turned into the Suns Hours which thing is co be done in this
manner.

Look upon the back- fide of the Quadrant for your Star, and lay the
Thred upon it; flipping the Bead down to the flope hours below, cill it
ftand upon the fame quarter and part (from fome juft hour on the left
hand of the Bead) with the Stars hour before found. Then note the
faid hour on che left hand which goech next before the Bead, for that
muft be fuppofed to reprefent che Stars hour, and muft therefore be
called by the fame mame or number that the Stars hour was. And the
following hours (from the Bead towards the right hand) mutt fuccef= .
fively take their numbers until you come to be under the day of your
Month. Unto which day if the Thred be daid, the Bead will (by
keeping of your former account )-{hew the crue hour, quarter, and part
of the Night. | s - |

Examplel, On Fanuary the 20th. the hour of Cor Leonis was ob-
ferved Eaftward of the Meridian, to be 9 and } part of a quarter. The

. Thred laid upon thar Stary on the back-fide of the Quadrant, will

crofs the flope hours.as doth the Line ‘AB, and the Bead put downto
che fore-mentioned parts of the hour, will ftand at the point B, So
chat the hour C mutt be called 9 a Clock, which is the obferved hour of *
the Star. Then the Line D muft becalled 10 a clock: and the Thred —
being put to January 20. (taken in the lower circular Line of Months) |
will lie in the line AE; andche Bead at E thews the time of the Night —
to be paft (the line D, that is paft) 10 a clock about 4 and 4 part of |
a quarter, whichis ts and § min. or 20min. paft ro at Night. Buc |
if this Obfervation had been made upon the fecond day of Nevensber:
then the Thred laid upon the day given in the lower Circle of Months, _
November 2, would licin cheline A F: and the Bead would be upon —
the fall Hour-line chat paffech through F,. which would be 4 a. Clock
in the morning, For if che line C beg, che line D is 10, che nexc line —
Is

ae Ay aE Merten Le non as soi ’ ;
"the Ofe of the Altimetrick Seale. es (33
314, and fo forward till your account fall apon Fs which muft be 4
-aclock paft {t 2 or) Midnight.. a ‘ { “hes.
' Example 11, Upon the 8 of Aaguf, the Star Aguila was feen on che
Welt fide of che Metidian, and the hour of it was found 3 and $ anhour
and 4 aquarter. The Thred therefore being laid upon that Star would
beasthe line A G, and: the Bead (reétified tothe 3 hourand } quarter)
would Randatthe point G. Sothat the next Hour-line on the left hand
of G, mult becalled 3 aclock, and theline F muft be 8 acleck.e Then,
| the Thred being removed to the day of your Month ( Augwf 8, in the
"upper circular tine of Months) will lie in the line A B; and the Bead
ac B will thew-the-Hourof the Night. Gf you-keep your former account)
to be & and half paft-£-a clock. Forif F be 8 aclock (asts before expref-
fed) chen che iafthour.of che limbis1t, the firftis #2, the fecond 1;
“beyond whichthe-Bead B. is about 22%, of an hour. _ Therefore the
hour of the Night is-t.avclock 22 sin.

. By thele Examples ‘the -manncr of the work will fufficiently appear in
all cafes, oe “i” 4

The Ufeof the Altimetrich Scale. .

HE Scale on the Fore-fide of the Quadrant nexe to the equal
i Limb is here called che Altimetrick Scale. It ‘is numbred by 1,
25-2, Ge. to 10, 20,30, &c. wooo, Each of which numbers are
beft fuppofed to be 100 fold, viz, 100) 200, He, to 1000, 2000, Ge, -
to rooeo: and alli thedefler parcs c{timated accordingly. The
_ ground on which you fland to.make your menfuration, is alfo fuppofed

‘

f to bea juft hey cian

I. Tofiad any Height at one Ol fervation.

“ Eryour Station be at E 5 and the fightsD A directed to the poine
L EF: the Thred.A B cutsoff the parts CB inthe meafuring Scale.:
which parts muftbe remembred. —Then meafure from your Station FE,
to the point H, which isjuftunder F. And always in this cafe: mulu-
ply this diftance EH by the fore- named parts of C B, and from the Pro-
dud cut off three figures coward therighthend. The Remainder is the
AltitudeGF. To which you muftadd HG, orD E, the height from

youreye at D to your foot aE.

.

Xxx Thas

ei.

Ne

162 Tb take tlsitndes sind’ Diftances: |
Thus-if the Thred AB thould cut of CB T¥OO parts; and the

Diftance E were 59 feet, the height G F would be 88; g000 or 78 4
feec. | ‘ a

te

II. To find part of an Altitude. —

Tee length of BX be only required. Scanding then at E, you
—/ may find the Altirude G F, Keep {till che fame Standing at E, and

find the Alcitude G X by che laft Precedent. So GF taken from G X,

gives FX required.

wl

IIL. Standing npon a known Height to find a Diftance.

ae the height FH be known, and the Diftance H K be required. Ore
der your {tanding fo, that the two fights P; $, the point B, and’

— the Diftance K, may allappear in one right line. Then look what den

grees the Plummet cuts off in the equal limb fromQ. Coune the fame
number in the fame limb from$; and there lay the Thred, as PT.
Note then what parts it cuts upon the meafuring Scale from Qto kK
Multiply chofe parts'into FH the known Altitude: and from the
Product cuc off three Figures the Remainder or Quotient is the Di-
flance MK, |

Thus

oh

Rae Pee PoE

e Eien Co tale By et ;

| = Totakedeferent Altitudes, = = 163
_ Fhusif che Thred PR thould cut off QR in the equal limb, 56 } de-
grees, the fame counted the other way from Sto T in the equal limb,
-and the Thred laid thereto would give 667 in the meafuring Scale.
‘Then FG being 88 4 feet and GH (fuppofe) five feet, FH mutt be
193% feet. This multiplied into 667, makes 62364: from whence
“cutting away the three right hand figures, there remains 62.364 or 62 *
‘feer for che Diftance H K.

IV. To find part of a Diffance.

FE che Diftance of K from Z were required. Firft, find HK, chen
BR AZ, by the chird precedent: their DifferenceisKZ. If KZ were
aTrench, you might from the Tower F, find che bredch of it withouc

any approach unto ite
V. To fied a Height at twe Obfervations.

} ih F H were to be meafured, and the way from Eto H were unpaffable,
DL fo char the Diflance of E from Hcould’nor be meafured. You
/muft in this cafe make two Obfervations. For which purpofe, take
your firft ftacionat E, and direét the fights D, A, to the point PB: no-
ting whar parts the Thred cuts upon the equal limb from CtoB. Then
po backwards ina right line, to a competent Diftance, asto M3; and
there making a fecond ftation, obferve (as before) what degrees the Thred
cutsupon the equal limb from No O: (the two fights L, I, being juft-
_ ly directed co the point F.) Then count thefe two Arks in the equal limb

from the contrary fide of the Quadrant, namely from D to Y, and
from L to # and applying the Thred thereto, look what parts it cuts
fromthe meafuring Scaleat Y and V. - Take the leffer number of parts
out of chegreater, noting the Difference. Meafure alfo the Diftance of
your two ftations, namely, from E to M, and add three ciphers to that
meafure; This laft number mutt (in this kind of work) be divided al-
ways by the fore-noted Difference s and the Quotient will give the Ali

—udeof F above G.
| Example.

“Ler the firtt Obfervation cut off 38 € gr. in the equal limb.- The
fecond 56} gr. Count the firft Ark from Dto Y: che Thred there.
aidgives 1250 in themeafuring Scale. The fecond fo counted from

LtoV, gives 667: The Difference of thele two is 583. Let the Di-
| Xxx 2 Rance

li

i

ies

number, with three ciphers added, is. 160000,
§83 (che former Difference) gives in the Quotient
for the Height GE. . And if GH be §,foot more =
HF willbe 93 2 feer.. |

- @ Note, That in chefe Menfurations, the point:

to ftand in the fame Level with the corner
DandL. Sothat GH, D E, L Mare all of
notetoo, that.the two ftarionary points are
thofle which are juftunder the corners D and

md 4

a tela ny eld mre ncaa i ben ae ener nen = Se ee eas

164. Totake-Alsitudes and Diflanves?.
{tance of the {tations meafured from E to M, be 51. 60 feet.

Which divided. bye
88.50. or 88.4 feer,

bye

The whole Heighth

G-is fuppofed to

of your Quadrant

one Heighth.
Eand M, namely,

FA

This

And .

le, ilies al
Ae ; +
sf

65?

ee oe

OF THE

yg , 1 oateivenceth
CANON,

| | ian 2 AN Dist 7 ras
Fables of LocariTHMs.

% Ogarithmetick is a Logical kind of Acithmetick, orarti-

| se . j
‘ i { (Mae ficial ule of Numnibers invented for the eafe of the Cal-
GP ASA culation, wherein each Number is ficted with an Artis
i we 02 4 ficial , and thefe Artificial Numbers: fo ordered, that’
ae he what is produced by Multiplication of natural Num-
MZ QO beys, the fame may.be effeGed by the Addition of thefe-

| their Artificial Numbers 5 what thcy perform by Divifion, the fame js

_ heredone by Subtraction : and fo che Hardeft part of Calculationavoi*
| déd by an eafie Profthapherefis.

'Albthis’ fall be made plain by applying that to thefe Arcificial Num-

bers, which I have fey down before, for the ufe of my Lines of Num--
"bers, Sines and. Tangents in. the Ulfe of the Se&or and ‘Crofs- Staff.
Wherein the Reader is co obferve, that what is to be wronghtby round*
Nambers only, is beft done by Mr. Brigges his Legarithms, but the —
Aftronomical part concerning Arks and Angles, by my Canon of Arti-

ficial Sines and Tangents. HAL (OE IO f |

. GHAP;

66: | The general Ufe of the. Canon
CHAP. I.

Concerning she Ufe of the Line of Numbers, Ihave fet-down ten
general Propofitions in the fir ft Book ofthe Ufe of the Croft.
Staff, Chap. VI. esd thofe may be applied to the Tabje of
Logarithms, ~ re . <q

; a i PROP.

nore at

To multiply one Number.by another,
His is the fixth Propofition of the ten s but I begin with the eafieft,

add the Logarithm of the Mulktiplicator,to che Logarithm ofthe Mul-
tiplicand, che Sumofiboth-thall beche Logarithm ofthe |Produd.

As when when wemultiply 25 ‘by 30; the’ Product is ~~" ~ 750
Sohere, add the Logarithm of 2 5, viz, 1.39794001
To the Logarithm of 30 1.47712125
The Sum of both will be : ; 2.87506126
And this is the Logarithm of 750, £
Inlike manner, if we multiply ro by 10, the Produd is 100
If too by to, the Produ@tisteoo | fo here ie
_, The Logarithm of 10 being 100080000
The Logarichm of 190 (hall be 2.06000000; —
ole 3»89000000,
10000 4:0coccoo dl
100000

5sCOCOD000

And foforward: All intermediate Numbers which have Intermediate

Logarithms. , | aah) |
Lfwemulciply toby ro, the Produ@js TOIO; of 102 by 10, the
Produ is 1020: |

fo,here
The Logarithm of 10, viz. 1.00000000
Added tothe Logarithm of ror 2.00432137
Gives the Logarithm of toro 3.00432137 _
The fame Logarithm of ro 1.00008000
Added to the Logarithm of rez 2.00860017
Gives the Logarithm of 1020 300860017

The

ef ey
<

<a Eg a :

ae “and Tables of Logarithms, -. "807
~The Differenice being only in the firft Figure, and that is always lefs
by onethanthe number of Places, in the Number given. As when we
find'the Logarithm tobe 2,00860017 the fit Figure 2 is Charaéte-
riftical, 7. e. che Index, thewing that the whole number 102 belonging
cothis Logarithm, confilts of three places. If the Logarithm had been
1,00860017, the whole Number muft have been 10.2 confifting of two
places, and che refta Fraction to.
If the Logarithm. were 0.00860017° the Number
belonging to it would be. 1, 02, 3. 1 and #2.. © And this.is
one of the reafons why the Differences were omitted in the firft hundred.
Logarithms. All chefe Logarithms may be found afterwards ander a
larger Index.

If weadd the Logarithm of § unto the Logarithm of 201, the Sum of
both fhallte the Logarithm of 1005, and the Sum of che Logarithms of
| sand 203 hall be the Logarithm of 1015. Thus che moft part of. che
_ Fable may be continued beyond 1000,

é PROP. Il.
| 4 Todivide one Naumaber by anothers

| Bs ast the Logarithm of the Divifor, out of the Logarithm of the

Dividend, the Remainder fhall be the Logarithm of the Quotient.

_ Aswhen we divide 750 by 25, the Quotient is 30 fo here
From the Logarithm of 750, viz, ; 2.89506126:
Subtract the Logarichm of 25, Is.39794001
There remains the Logarithm of 30 1.4771 212g:
In like manner, when we divide 11 by 4, the Quotient is 24, fo
‘here che Bogarichm of 4, viz. 0.60205999.
— Taken fromthe Logarihm of 14 1.04139269.
| sia
Leaves the Logarithm of 2 4 0. 43933270

Wherefore, if ic were required co find che Logarithm of a whole Num-

| ber with a Praction annexed (as of 24) we might firft reduce itinto an:

improper Fraétion of *4 (or racher of 723.) and chen fubcract as before.
If ic were required to find the Logarithm of a fingle Fraction, .as-

of 4; » we may fubcract as before; Butthis Fragtion being lefs than hig
7 the.

|
|
\

Again, if we multiply 201 by ¢, the Produé is roog§: fo here = :

7 -

SO SS an? OF OAS le gee Orgs eae Paw E
- x . 4

368, Thegeneral Uftof she Catun

the Logarithm muft be le(s chan 0, and therefore noted with — a de- |

feGtive fign. | a REY TRS:

_ So the Logarichm of 44 or 24 hac haces | Sa vaen
And che Logarichin of fq 0 eS 0.4393 527

Lich abe 95 oh et 9 Su lapiciic

_...Tofind the Square of a Number.

die
ed
4
7
a i

“WAL che Logarichm of che’ Number given is the full Logarithm |

the Square Root.
~ “So.the Logarithm of 144 being
The half chereof is ie RH
che Logarithm of 12, and fuch is the Square Root of T44,

2.15836249 |
— T.07918124

Then by converfion, having extra&ted the Square Root, we may foon

find the Logarithm.
As the Logarithin of to,o000 being
The Logarithm of the Square Root 316227, is
And for the Root of that 177827

PROP. IV.
To find the Cubique Root of 4 Number.

Logarithm of the Cubique Root.
Sothe Logarithm of 125 is
And}the Logarithm of 5 ail 0.698 97000
By the fame reafon we may find the Biquidrate Root, by dividing the

Logarithm of the Number given by 4: the folid Roor, by dividing by 5,
and fo forward, - . ee

And by converfion, having extracted the Root, we may (oo find the

Logarithm.
As the Logarithm of 10.000, éc. is

2,0969100L

1.00000009
0-50000000
0.2 500000gmy

all

a
f Be third pare of che Logarichm of the Number given. isthe fall

- 1.00090900
The Logarithm of rhe Cubique Root, 29544 9033333333
The Logarithm of 100.000, &e. 2.00000000
The Logarithm of the Cabique Root 4641 0.65666666.

Then multiplying che Squareand Cubique Roots one by another, we may

Produce infinice ocher Numbers, and have all their Logarithms. .

PROP,

i Three Numbers Veing given, to find a fourth
| PI His Golden Rule the meft ufeful of all others may be wrought f{e~-

-| number here required.

.

1]

|

|
:

<i

‘|
Bi

| tL vera

|

Product by the

|

the Logarithm of

i

i

| The ordinary way in

Sie a Pe eee Be amis EL ee Bes NL ae hy

: 169

+ and Tables of Legarithms.
VeOPROR, Vues

«

ss
ae

Proportional.

lways, as it appears by this Example: :
4 to a fourth number.

Arithmetick is by Multiplication and Divifion.
For firft they multiply che fecond into the third, and chen divide the
firtt Number given. Ashere, mulciplying 24 by 4, the
hen dividing 96 by 12, the Quotient will be 8, the fourth

A; 12 wnto 243 fo

Product is 96, t

According to this way weadd the Logarithms of the fecond and third,
d fubtra& the Logarithms of the firft, fo thac which remaineth thall be

an
the fourth Number required.

. Thus che Logarithm of the firft Number 12 is - 1.07918135
The Logarithm of the fecond 24 1 38021134

~ The Logarithm of thethird 4 0.60205999
|) ‘TheSum of the fecond and third Logarithms --¥.98227123
Subtract the firft, and there remaineth @ 90338998

the fourth Proportional.
k is by Divifion and Multiplication. For
where thefecond Number is greater than the firft, they may divide the
| d then multiply the third by the Quotient. As
‘chen multiplying 4 by 2,

And thisis the Logarithm of 8,
A {econd way in Arithmetic

Tecond by the firft, an
here, dividing 24 by 12, the Quotient is 2:
the Product will be 8.

According to this way we taket

he Logarithm of che firft out of the Lo-

Ueerithm of che fecond,and then add the difference to the Logarithm of the
third. So the Sam of this Addition (hall be che Logarithm of che fourth
required. — "resit, : Re
| Thus the Logarithm of the firft Number 12 1s LO79I8T25 -
The Logarithm of the fecond 24 1.38021124
| _ The Difference berween the increafing _ 30107999
Added tothe ‘Logarithm of 4 0, 60205999
|| Gives che Logarishm of 8 0,90308998
Bee ; Yyy | A
‘ we)

Kuo

Tadus 2
iVic

{us per I.

ed,

uotiens
2 per a di-
vif mul-
tiplicatus

in tertits

170 The general ve of the Canon | ae.
Ill. Athird way in Arichmetick is by Divifion and Divifion, for where.
Qrotiens the fecond Number is lefsthan the firft, they may divide the firft by
4 pet 1.0 the fecond, and chen again divide the third by che Quotient. As,
diyifor 3, here, dividing 13 by 4, the Quotient is 3 ¢. thendividing 24 by 3, the
Quotient is 8, | ae,
According to this way wetake the Logarichm ofthe fecond out of the. |
Logarithm of the firft, and. then take the Difference out ef the Logarithm,

of thethird: fothat which remaineth fhall be the Logarithm of the fourth.
Number required. |

‘Thus the Logarithm of the firft\Nimber 42: is. 1.07918125.

The Logarithm of the {econd 4. ©,6920§999_
The Difference decreafing- | 477121265
Subtraéted from the Logarithm of 24 1.38021 124,
Gives the Logarithm of 8. 0.90308999:

Thefe two latter ways by Difference of Logarithms, may be confider--
edasthefame. Though there.befome difference between them, yet thas
may eafily be reconciled, if we have regard to the nature of the queftions.
For three numbers being given.in. direct Proportion, if the fecond be-
greacer than the firft,che fourth muft be greater than the third: Ifthe fecond::
be lefs than the firft, che fourth muft be tefschan the third,and their Loga-
rithms accordingly. But in reciprocal proportion, confidering the fiftand:

~__ tecond numbers tobe of ene denomination, we are to obferve the contrarys.
- Iftwe defire to turn SubtraQion.into Addition, we may take che Lo-
garithm which isto be fubera&ted out of the Radius, and add the Com--
plement. So the Sum of this Addition, the Radius being fubtracted, thal.
give the required Logarithm as before,
Thus in the laft Example : where fubtra€ting the Difference 4.7712126 5

outof 3.38021124, the Logarichm of 24, we found the Remainderto be.
, ©,90308998 the Logarithm of 8,

The Radius being | 10.00000G00.
Khe Logarithms to be fubrra@ed : | 047712126
~ "The Complement to the Radiusis « 9.52287874,
This added to the Logarithm of 24 138021124,
Gives us a compound Logarithm . - 10,90308998: |

From.

li
}
\!

“

.
— =

Se ear Oa oe + ee

; a ‘and Tables of Logavithzs, TY
9} From this, if we fuberact che Radius, (that is, if we cancel. che firft
| figure co the left hand) the reft is 0.90308998
the Logarithm of 8, the fourth Proportional, as before. m
| By help of this fourth Proportional we may come fomewhat near to
find a Logarithm for a number of 6 places. |
As ific were required to find a Logarithm for this number 868624,
the Table will afford us Logarithms for a leffer and a greater number 5
and then che intermediate may be found by the part proportional in this

manner. :
Here we have the Logarithm of 868 2.9385 1977

ties

And the Logarithm of the next following 869 2.93901978
And the tabular Difference between chem | 50005
Tf che Index be fitted to the number of places,
The Logarithm of 868000 _fhall ve §.93851972
| And the Logarithm of 869000 5-939019978
| The Difference being 1000 §0005

_ Then taking 868000 ope of 868624, (the number given) the third
_ Difference willbe 624. And having thefe three Differences the Pro-

>

- portion will hold.

me - 4. 1000 ants 50005
i So - 624. axto 31203 the part propor-
“tional to be added to the leffer Logarithm 5.93851973

fo thall we have 5.93883176 for the Logarithm required.
Inlike manner, having a Logarithm given, we may find the value
- of it ina number of fix places.

As if the Logarithm given were 7-93883182
_ and ic were required to find the Number to which it befongeth :
This Logarithm is not to be found in the Table; bur changing the

Index and making it 2 93833182

The nexe leffer Logarithm of 868 is 2.9 3851973
And the tabular Difference following 50005
) And the proper Difference 31209 -
Asthetabular Difference 50005 vento 100600
_ So the proper Difference 31209 ante 6241

_ the part proportional to be joyned to theend of che former number 868 ¢
ih Yyy 2 fo

|

~

Me of

172 The general vufe of the Canon

“fo thall we have 86862411 for the value of this Logarithm. But the-

Index of the Logarithm being 3, the Number required muft confift- of -

four places, viz. 8686, and che reft'a Fraction of ine.

s

This I fay is fomewhat nearthe Truth. For'this number here propos

fed 868634 is the Square of 932,

The true Logarithm of che Root 932 is 2696941591:
The true Logarithm of the Square 868624. 593883182.
PROP. VI. |

Three Numbers being given, to find 4 fourth ina duplicated
Proportion. ie *)
: TN Queftions that hold in a duplicated Proportion between Lines and ¢
J Superficies, the Logarithms for Lines given may be doubled, the Loe.
garithms for Lines required may be halfed, and then the work will be.
the fame asin the firft part of the former Propofition.
Suppofe, the Diameter being 14, thecontent of t
the Diameter being 28,. what may the content be? . °

he Circle was. 1 54,

Here the Queftion concerns both Lines and Superficies, I double
the Logarithms of thetwo Lines given, and then work as before in this

=

|

|

|

manner : I
The Logarithm of ¥4 ‘is T.14612803 |
The Logarithm of | 28 1.447 15802
The fame again 1.44715803 |
The Logarithm of 15.4. 2618752072 »
The Sum of chefe laft . 5-08 183678
Subtraé che double of che firft 2.29225606
There remainsthe Logarithm of 616 -

And {ach ‘is the content of the Circle here required.

: Suppofe the content of a Citcle being 154, the Diameter of if was 14 § \

the content being 616, what may the Diamerer be ?

Here: being one Linegiven, and one Line required, I-double the Lo- |
garithm of che Line given, and then working as before, the half of the.

remainder {hall be the Logarithm of the Line requireds.

2.78958072

Thus. |

Beal
op RR

and Tables of Legavitlis,
‘Thus the Logarichm of 154 is ;

573

2.1895 2072

| tent required.

The Logarithm of 28 the Diameter required.

Or according to the fecond manner of operation, the difference be-
cween the Logarithms of Lines given may be doubled ; the difference be-
tween the Logarithms of thecontent given may be halfed, and then the
work will be the fame as inthe latter part of che former Propofiticn.

So in the firft Queftion, where the Diameters were given and the con-

The Logarithm of 616 — 2,98958072-
The Logarithm of 4: 1,14612803

" The fame again» 1,14612803
The Sum of thefe three laft : : 5.03183673_
Subtra&t the Logarichm: of the firft” 2.18752072
The Remainder will be 2.80431606
The half thereof is - 154471 5803

v

The Logarithm of 14 is - 1,94612803 »
The Logarithm of 28: 144715802
The Difference increafing . 30103600
The double of this Difference 60 G6 bem
Added tothe Logarithm of 154. 2,18752072
Givesthe Logarithmof _- 616 2.789 58072

Inthe fecond Queftion, where the content of both the Circles wasknown,
and the Diameter of the one required.

2.18952072 -

The Logarithm of 154:
The Logarithm of «: 616 2,.78958072 :
- The Difference increafing » 6®206000 »
The half of this Difference: "30103000 »
Added tothe Logarithmof 14+ 1314612803 ©
( Gives the Logarithm: of | 23. 1,4471 580 z°
PROP, »

BT PUP MN YAP Aa eye CU, TN ra eR ReRRtR ep Seg Leet nO ae eS PE OCT MOERN ie ee
rae ek esta x iy ‘ : Ba Ae ‘ i
y

—

fa

374 Thegeneral Ufe of the Camm = =:

re

PROP, VII.

Three Numbers being given, to fied afourth in a triplicated
Proportion. | |

FN Queftions concerning Proportion between Lines and Solids, the
& Logarithms for lines given may betripled; the Logarithms for lines —
yequired may be divided into three parts, and then the work will be the
fame, asin che firft way for che Rule of Three, 3

Suppote the Diameter of an Tron Ballet, being four inches, the weight
of it wasnine pound, the Diameter being eight inches, what may the
weight be? - i

The Logarithm of 4 is 0.60205999
The Logarithm of 8 0,90308999
The Triple of it 2.70926997
The Logarithm of 9 ©.9542425E
The Sam of thefe laft 3.66351347 |
‘Subtraé che Triple of the firft Logarichm 1530617997

There remains the Logarithm of 72 185733251

; And fuch isthe Weight required.

Suppofe the Weight of an Iron Bullet being nine pound, the Dia-
“meter was four inches; the Weight being feventy two pound, what
may the Diameter be ?

The Logarithm of 9 is ©.95424258
The Logarithm of 72 1.85733250
The Logarithm of ©,60205999
The double of chis again 1.20411998
~The {um of thefe laft 3-66351247 |
The firft Logarithm fubtra&ed, there remains 2.70926996

The third part chereof is

| 0.90308999
The Logarithm of 8, and fuch is the Diameter required. ss |

Or

7 aes . wer ee ON oe ee Re | ee tr ee OS By Se

and Tables of Logarithmse- 15
Or according unto the fecond manner of operation in the Rule of
Three, the Difference between Logarithms of lines given may be tripled ;
the Difference between the Logarithms of the Solidiry. or Weight given.
| may be divided into three parts,. eth |
| So inthe firft Queftion,. where the. Diameters were. known, and the
Weight required. |

The Logarichm. of aa 15, 3 0.60208999:
The Logarithm of : 3. re) 90308999.
| The Difference increafing 30103000
: The triple of this Difference. : 90309000
i Added tothe Logarithm of = 9. .9$43425 4:
| Gives the Logarithm of. 72 1.85733258°
Inthe fecond Queftion, where the Weight was known, and the Di-
* ameter required.
| The Logarithm of = 71. 1.85733250:
| ee
| The Difference increafing : 90308999:
The third part of this Difterence 30102990
Added to the Logarithm of: 4 ©.60205999°
Gives the Logarithm of : | Gh ©.90308998

PROP. VIII.

Having two Numbers given, to find athird in continnal Proportion, a fourth, |
afifth, afixth, and fo forward. |

a py Coorcins tothe firft way in che Rule of Three, we may {ubtrac-
LX. the Logarichm of the firft number, out of double the Logarithm |
ef the fecond, the remainder thall be the Logarithm of the third, then.
fubtracting the Logarithm of the firft Number again, out of the Loga-.
rithms of the. fecond and third, that is, out of triple the Logarithm.
of the fecond, the.xemainder fhallibe che Logarithm of the fourth, and’

fo forward. .
;" Ass,

|

ot
© NS eat

196 ~The general U[e of the Cams

>

As, when we fay: As tanto 2, fo 2 “nto 4, and.4 unto, and &
unto 16, Sc. Becaufe the firft Number is r, there is no need of Di- ©
vifion, bue only co multiply 2 the.fecond Number into it felf, the Pro-~
du& gives the chird Proportional Number to te 4.: then multiplying 2
into.4, the fourth Propertional is $: and multiplying 3 into 8, the —
fifth Proportional is 16; and fo forward. So here the Logarithm of

che firft number being 1, there is no need of Subtraction,
But finding the Logarithm of 2 to ‘be
The double gives the Logarithm of 4
The.Triple gives the Logarithm of 8
The Quadruple give the Logarithm of 16
— andfo forward ininfinitum.

In all other numbers that begin nor with 1, we may either fubtract

0.90308999

K,20411998 ©

whe Logarithm of the firft Number or.add the Complement unto the

Radius.
As when the Numbers given are 100 and 1068.

The Logarithm of the firft Number 100 is
The Legarithm of the fecond 108
From rhe double of this fecond Logarithmg,
Subrraét the firft Logarithm, «there remains

_the Logarithm of 116 &* the third Proportional. _
Again, fubcra& the firft Logarithm
Out of the Sum of the Logarithms of
The fegeond Number and the third Proportional
There remains the Logarichma |

2.Q00000000

2.03342276

4.06684752

2,06684752

t!
yy
i
}
|

2.00000600 ©

5
bal

-0.30102999 >
©.60235999

me

|
ny fi

2.03342376 |

2.06684752 _
2.09927128

an{wering unto 125 *2* the fourth Numberin continual Proportion, |
According to the fecond masner of Operation we may take the Dif-
ference between the Logarithms of the two Numbers givens fo this Dife
_ ference applied to the Logarithm of the fecond Number, thall give che —
Logarithm of che third Proportional: the fame Difference applied to
‘Logarithm of the third Proportional, thall give che Logarithm of the fourth
Proportional, or the double of this Difference applied tothe Logarithm
of che firft Number, hall give the Logarithm of the third Proportional ¢
the treble of this Difterence applied to che Logarithm of the firft Number,
fhall give che Logarichm of the fourth Proportional; and fo forward.
As inthe former Example where the two Numbers given were 100 and ©
108, fuppofe 100 increafing to 108, and f> yearly in continual Propor- _
tion after the Rate of 8 in the 100, and that it were required to find

what this 100 would grow unto by the end of 20 years,

/

The

|

.
e-

-

<< -

-

ee EN er a es een oe Py ne
_~ \ ’

ie 2 The Logarichm of the firft Nunaber-100 is 3.00000000
The Logarithm of the fecond ~~ ‘108 2.03342376
The ycarly difference increafing 3342376

— Geeeecareenomemt ce

Added to the Logarithm of the fecond, gives 2.0668 2752
“the Logarithm of 116 +4* for the third Proportional ; And-fuch is the
“increafe at the end of che fecond year.

Proportional, gives 2,10025128

the Logarithm of 125 *2+ forthe fourch Proportional, and the increafe
atthe end of the Hird year, and fo the reft. 1

Bur becaule the Queftion is only of the 20th, year without knowing

the reft,we may multiply the former yearly Difference 3242376

(Qe, eens Se

and Tables of Logarithms; = 179

Again, the fame yearly Difference added to the Logarithm of che third

| By 20: {othe Difference of 20 years ~ 66847520 |

_ Added to the Logarithm of the firftt Number 100,viz. 200000000
___ Gives the Logarithm of 466 °2* 266847520
thacis 466/. 1s, 114. fere, theSum that 100 would grow unto by the
end of 20 years atthe rate propofed, -

. Inlike manner af che two firft Numbers given were 108 and 100:
Suppofe 208 decreafing to the 109) and fo yearly in continual proportion
“and thar it were required to find what 10@ would decreafe unto by the
end of 20 years: Or (which is all one) {appofe 100 to be due 20 years
hencey and thar it were required to find the worth chereof in ready mo-

ney according to the former rate.

The Logarithm of the fift Number 108 is 2.03342376
The Logarichm of the fecond 100 2 00Qe0000
The Differences for the year decreafing 3342376
— Taken from the Logarithm of 100 leaves 1.96657624

‘the Logarithm of 92 4° for the third Proportional, and fuch is the
“prefent worch of 100/. dueat che years end.
* The fame difference fubcracted once more leaves 1.93315248
The Logarithm of 85 24 for che fourth Proportional, and the pre-
‘fent worth of 100 /, dueat the end of two years.
The fame Difference mulciplied by 20 makes 66847520
And {abtracted from che Logarithm of 100, leaves 1.33152480
“the Logarithm of 21 4548 chacis 211. 9s, a4, and fuch is the prefenc

LEZ worth

ms) Pet a in Soe a KC aN wetee UR a AH ay e

i The general Ufe of the Canon : :
worth of 100 7, duear theend of 20 years; So that this ‘prefent worth.
being taken forth of the too /, principal debr,there remains 78/. Lost Id,
for the prefent worth of che continued gain that may be made either of

theloan of r00/, orof 8/, Annuity after 20 years according to the for- |
mer rate, : 3

Ifa Leafe. of 100/, by the year, or fach other yearly Penfion were.
tocontinue for 20 years and that it were required to find the worth
the worth thereof in ready money. This might be found upon the fame
ground of continual proportion, and that feveral ways. +

| |

1. Tcappeareth before, that roo/. dueat the years end is worth bye
92 *°2in ready money : If itbe due at the end of two years, the pres’
fent worth is 85/, 222: then adding thefe two together, we haye

1781], 224 for the prefent worth of 100/, Annuity for two years, and
fo forward. ; !

2. It appeareth before that the prefent worth of 8 /. Annuity for 20
years is 78). §452: and then it follows by proportion, | |

|

As an Annuity of | $1. 0000 0.90 308099
/s tothe worth thereof 78.5452 1689511953
2901

Se an Annuity of 10©.0006 2,00000000
Unto the worth thereof 981.8147 299203954

:
3+ As the yearly Loan of r00/, includes an Annuity of 8/. So there
is 2 Sum equivalent to 100 /. Annuit

his Sum equivalent may be diminifhed according to the Number |
of yearsas before: to che Com plement of the Sum diminithed to the Sum |

|
,
|

-
1

. °
equivalent fhall be che prefent worth of the Annuity,. |
As the yearly gain of § 0.90308999

To the Loan of 100 - \2,00000000 |

Soan Annnity of | Kole 2,.0000000@

To the Sum equivalent 1250 3.09691001 |

Then |

Fs |

B and Tables of Logarithms. to
| Then for diminithing of this Sum equivalent, we may multiply the ©
— 3342376

_- By 205’ fo the Difference for 20 years 66847 520
Taken from the Logarithm of 1250 309691001
There remains the Logarithm of 268.1853 ~ 2.42843481

Whofe Complement to £250 is 981.8147, thacis 9811. 164.3 d, 9b.
and fuch is the prefent worth of too /, Annuity for 20 years, at the

arate of 8 inthe 100 per annnw. bis 7 |
_ The like reafon holdeth for any other rate and time propoled.

ye PROP. IX.

i

. Having two extreme Numbers gives, to find a mean Prepor-
tional between them. ae

OA Dd the Logarithms of the two extreme Numbers: the one halt >

\¢ of the Sum {hall be the Logarithm of the mean Proportional.

Asif the two extreme Numbers given were 8 and 33.

The Logarithmof 8 is 0.90308999

The Logarithm of 32. | 1.50514998

\ The Sum of both Logarithms 240823997
1.20411998

The Logarithms of 16: and {uch is the mean Proportional here re-

i
, t

: The half of this Sum 1s .
| quired,
F PROP, X.

Having two extreme Nambers given, to find two mean Pro-
_- portionals between them.

‘ ie
EN the ordinary way of Arithmetick we commonly multiply the-great-
|B ex Extreme by the Square of the leffer, fo the Cubique Root of the

|
| ProduG (hall be theleffer mean: then multiplying the lefler Mean into

_ the greater Extreme, the Square Root of the Product fhall be the greater

| Mean Proportional: Or having tound the lefler Mean, we may find the

other Mean by continual Proportion. — |

| Accordingly we may add the Logarithm of the greater Extreme, to

| double the Logarithm of theleffer, fo the chird part of theSum (hall be

| ¥ the

}

LEE2

: 4 ys AS eo SHEA) ee " ‘

180 The general Ufe of the Canon ‘
the Logarithm of the lefler Mean. Then adding chis Logarichm of |
the leffer Mean, to the Logarcehm of the greaccr Extreme, the one half
of the Sum-thall be che Logarichm of the greacer Mean Proportional,.

Asif che two extreme Numbers given were Sand 27.

Add to the Logarithm of 8, viz, 0.90308999
The fame again | 0.903 08999
And the Logarichm of 29 1.43754374.
The Sum of thefe will be 3-2375§4374,
The third part of this Sum is 1107918125.
the Logarithm of 42 the lefler Mean Proportional, }
Add to this Logarithm of the lefler Mean 1,07918123
The Logarithm of the gr eater Extreme 1.431 36736

The Sum of both Logarichm will be 2.§1054501

And the half of this Sum is 2.1 5527250
The Logarithm of 18, the greater of the two Mean Proportionals here
required. |

Or according to the fecond manner of Operation in the Rule of Three,
(which is the work that I always follow in the line of Numbers) we
may take the Difference between the Logarithms of the two extreme |
Numbers, and divide this Difference into three equal parts, fo the Sym.
_ of the Logarithm of the leffer Extreme and 3 part, fhall be the Loga=
rithm of the leffer Mean: the Sum of this Logarithm of the lefler

Mean and the fame 4 part, fhall be che Logarichm of the Greater Mean.
Proportional, , |

So the Logarithm of 8 being

©.9030900:
The Logarithm of 27

164313637
$2827 37 |
1760912

1.0791812 —

The Difference between them
The third part of this Difference
Added to the Logarithm of 8 gives
the Logarithm of 12 the lefler Mean,
The fame added to the Logarithm of 12, gives.

| 2552925
che Logarithm of 18 the Greater Mean Proportional.

Unt

And |

NB 9 se oi

and Tables of Logarithms. — 181
And by the fame reafon, if it were required to find three Mean Pro-
portionals, we might divide che former Differemce into four equal parts

and fo forward.
Asif ic were required to find che firft of eleven Mean Proportionals

between rooand 108, Or(whichisall one) fuppofing too/, increafing

_incontinual Proportion, foas thae by the end of £2 months it: came to

roi. and thar ic were required to find what this t00/, did. grow
unto by the end of che firt Monch.

The Logarithm of the firft Extreme 100 is 3,0000000
The Logarichm of the fecond 108. 2.0334237
The yearly Difference between them | 334237
The 12, pare or monthly Difference 27853
Added to the Logarithm of _1€0 gives 3.0027053

The Logarithm of 100.6434@301 the firft of the eleven Mean Propor-
tionals: and the growth required. |

Then having thefe two, roo and 100.64340301, together wich
108, thelaft of twelve, the other Intermediate may be found by con:
tinual Proportion as before. Ds

This Explication of my tem former Propofitions: may. ferve fon he
frugal Ule of the Table of Logarithms. Thofe which require more may
have recourfeto that Treatife which is mentioned before in the Front of

the Table...

182 Pee, ee general U[e of the Canon

CHAP. Il.

Concerning the Ufe of the Lines of Sines and Tangents in the

refolving of Spherical Triangles.

fhewed in general, in the feventh and eighth Chapters of the
firft Book of the (70/?-ftaff, how they might ferve for the Re=
folution of all Spherical Triangles. More particularly in thé Ulfe of my
Seftor, Chap. §. reduced thac which is commonly required ina Sphe-
rical Triangle into 28 Cafes. And for thefe they may be all refolved by

CO test the Ufe of the Lines of Sines and Tangests I have )

ty Tables of Artificial Sines and Tangents without the help of Secants or

verfed Sines.

Thismanner of the work will be always fuchas in the ordinary Rule
of Three, For, here we have three Numbers given, whereby to find a
fourth Proportional, And therefore either we may add the Logarithms
of the fecond andthird, and fubtra& the Logarithm of che firft : |

Or we may take the Difference between the Logarithms of the firft
and fecond, and apply that Difference to the Logarithm of the third.

The firft of thefe ways is beft for the refolution of right angled Tri-

angles where the Radius, viz. 16000000 is one of the three Num-

bers given, but the fecond way by Differences is more convenient for
thereft. - *

The like manner of work may be obferved when we are to confider

the Sines or Tangents of Degrees, Minutes and Seconds. For the Seconds, _ |
not expreffed inthe Canon, will be found by the Part Proportional: as I

will fhew in the Examplesfollowing.

1. Ifit were required to find the Sine of 51 gr. 32 min. 16 [ec. I
fhould find, : ; a

The Sine of $1deg.32™, is 903937452
The Sine of $ deg. 33 m. 9:8933455§
The Tabular Difference between them 1003

Then the Difference becween 32 wm, and 33 m. being 60 Seconds, the Pro= _

portion will hold,

04s

1%
(a

ee ela ete a ee. me ee * “Rare YY ee ey ay
‘i >) Ae ae Fy S. pene teary ee ee Geet eas oe ewe ee Sie

A; 69 Seconds ante | 1003

So 1 unto ast the part Propertio- ~

» ae
nal to be added unto the Sine 51 deg. 32 min.
So thall weave 9.8937703, for the Sine of 51 deg. 32 min. V5 fee.

2, If it were required to find the Degrees, Minutes and Seconds be-

~ longing to this Tangent 10 ©999782

I thould find by the Canon that this is fomewhat more then the Tan-

| gent of 51 deg. 32 min. | —-:10.09991 34
. Lefsthan the Tangent of 52 deg. 33 min. 10,1001 728
The tabular Difference between thefe is 2594

648:

And the proper Difference is |
between the leffer of thele Tangents, and the Tangent given: there-

fore,
As 2594 unto 60 Seconds.
So 648 unto + R504 And fo, I find

4
this to be the Tangent of 51 deg. 32min. 15 fee.

3. If ic were required to find the Sine belonging to this Tangent
70.09997982, I thould find the Ark to be fomewhat more than $127.
31 min. and the Sine correfpondent fomewhat more than 9.893745 2
then taking out che Differences as before, I find, that |

As the tabular Difference of Tangent 2§94 3,4139700

648

Is to the preper Difference 2.8119750
6023950

So the tabular Diffircnce of Sines 1003 . | 3.0013009

To the Part proportional 25 E 2.3989059

| This Pare proportional added unto the former Sine: 9:8937452

gives 98937703 for the Sine required. .

Thefe Premiffesconfidered, Icome to the 28 Cafes before-mentioned,

wherein I fer down a Canon and an Example for each Cafe, and thefe

for the moft part the fame which Tufed before. |
Thofe which have no further ufe but of Degrees and Minutes, may

“take thar Sine or Tangent which they find to be next in the Canon, and

neglect che Seconds. | :
| In,

and Tables of Logavithenss Hiie, —. 382 |

184 iG: The general Vie of the Canon
Taa RECTANGLE TRIANGLE,

1. To find a Side by knowing the Bafe and the Angle oppofite
" to the enguired Side.

As in the Re&angle Triangle A C B,
wherein A ftands for the Equinoétial point ;
A B, an Ark of the Ecliprick reprefenting
the-Longitude of che Sun in the beginninf
of 3, BC an Ark of the Declination og
the Sun from the Equator, and AC an
- Ark of che Equator reprefenting the Right
Afcenfion of the Sun in B: Knowing the Fi
Bafe ABto be 30 gr. and the Angle
: is C 23 gr. 31 min. 30fec. if it were required to find the Side

- da om. fee. /
As the Radius, the Sine of 9° ©0 ©0 10.0€06000
Is to the Sine of the Bafe 30 08 oO 9.698 9700
So the Sine of the oppofite Angle 23 31 30 9.6011 352

Tothe Sine of the Sidereguired 1 30 43 19.3001052
And fo writing the Sine 9.60010§2 in-a Paper by itfelf and hold-
ing to the Sine of the Bafe inthe Canon 1 gr. 2.3. 4.5. and fo for-
ward, it would be no long work co write the Sum in a Column by ir-
felf, and fo find the Declination for each Degree and Minute ef che
Ecliptick. ‘
. 2, To find a Side by knowing the Bafe and the other Side.

Asin the Rectangle ACB having AB 30 gr. andBC 11 gr. 30 ms,
43 fec. to find the Side AC. |

As the Cofine of the Side given It 30 43 99911740
Isto the Radius | 90 ©0O ‘00 10,000C000

So the Cofine of the Bafe 30 ©0 oO 9.9375 306
Te the Cofine of the Side required 27 §3 «43 “ 2. 946 3566

3+ To

; aah eae
ores

gna Tables of Logarithms, . | 185

3. To find a Side by kuoving she too Oblique Angles

Asin the ReGangle A C B having CAB for the firft Angle . gr.
31 min. 30fec. and ABC for the fecond 69 gr. 20 m, 3§ fec. to find
the fide AC.

| As the Sine of thenext Angle = 23-3 30 9.601352
Is to the Radius 90 00 CO 10,cO0KCeCee

So the Cofine of the oppofite Angle 69 20 35 9.5474918 -

To the Cofine of the Side required 27. «$3. «43 9.9463566
| 4 4. To find the Bafe by kwowing both the Sides.

| As inthe Rectangle ACB, having A C 27 gr. $3 8. 43 fec. and
BC, 11gr. 30m. 43 fee. to find the Bafe A B.

Asthe Radius 90 ©® CO 10,00900@e0-
| ae

a To the Cofine of the one Side 27 53 43 9.94635695
| $o the Cofine of the ether Side The 39 43. 9 99T1640
i; To the Cofine of the Bafe 30 0@ CO 9.9375306

/ 5. To find the Bafe by knowing one Side and the Angle

~ oppofite to that Side. ;
As if-in the former Triangle AC B we draw BD and Ark of the
“Horizon for the Latitude of s1gr. 30 sin. reputing the Ampliude of
ethe Suns Rifing from the Eaft, we fhall have two Triangles more, one
Rectangle BC D, the other Obliquadranglei A BD, and fo in the
“Restangle DC B, having BC 11 gr. 30m. 43 fece and BDC 38 gr.

Zomin, ifit were required to find the BaleD B. -

As the Sine of the Angle 38. 30 90 9.7941495 ©
| | To the Sine of the Side 11.430: 43 9.3601052 _
_ Soss the Radius ee go 00 00 10,0000000
To the Sine of the Bafe . 18 4t. 56 945059556
. Aaaa 6. To

| 3

} -
>

| PATE REE. <a Rea - eee ee) at 6 cht ee bay ee 2s 4 ers. tee
1S is SE ioe: , hg Td Pes Ed ae Ee TIE sok Uh

We ae ct FA er eee,

* ' Pe. < 0 a 4

fits

186 |
6. To find an Angle by knowing the other. OS ligne Angle, and the ,

As in the Re@angle AC B, having BAC 23 9. 31 min.

and AC 27 gr. 53 min. 43 fec. to find the Angle ABC..

As the

att pedi ia. UR Ea eB Ss Ad fails oat aide Bomar ality | ik ae ae re 2
vue are eh) edie GIS) hoe i teak My Te Oe asi seta aN pacer IN
. ste) ‘ Piette tai ilies it

Nips

The gentral Ufe of the Canon —

. SALA bia
op aa ;
as +o WV oe ‘

Side oppofite to the Angle required.

Radius 90

To the Sine of the Angle Liven 33

Sothe Coftne of the Side 27

To the Cofine of the Angle required 69

ule. ne an Angle by knowing the other Obligue Angle, and the
Side opposite to the Angle given.

As inthe Re@tangle A C B, having BA C 23 er. 31 min
and BC 11deg. 30 min. 43/ec. tofind the Angle A BC,

As the Cofine of the Side Pal

ME ,
' Tothe Cofine of the Angle given 23
Sos the Radiv 92 oO

Tothe Sine of the Angle required

8. To find an eAngle by knowing the Bafe, and the Side
oppofite to the Angle required,

As in the Re@angle BC Dy, having BD 18 gr, 41
BC i1¢r. 30 min. 43 fee.

es the Sine of the Bafé

fs to the Radius .
So.the Sine of the sppofite Side I

To the Sine of the Angle
Thefe eight Propofitions have been wrougl

following

require yoyne help of Tangents,

69.20. 35°. 9.97174

to find the Angle BDC, |
18 41 56. 9g 5050000

90 ©0 00 10.0000C00

38 30 00 9:7941495_

30 fee.
oOo 0O 10 000C090 |
Hi ie Ps, | 9.6011352_
53 43 9.946 3566

20 35 “19.5474918m

+ 30 fees
t 30 43 9.911740
33% 30 — 9.9623153

CO. 10.0000000

: -

m. 56 fec, ands

ey,

I 39 43. 93001052.)

ae,

it by Sines alone 3 the eight |

. To

¥ ray nt) f es rae! h ¢) etree Res

wae Uta ci Mian Aa A eb OCR a AS Sie aR ee ae a
r eee o! SWAT Fae Tes eS MM KR AE aks

ve i 4 d a
Mu 5 ; ae a : er +" she Ne hz
4 yd Tables of Logarithms. —
ss I P. i ¥ . oh st KE o

ana Tables of. gare WINS

rad: adi ite to the Side neidibiy

‘2B AC 23,gr. 31 min Boi/ees to find the Side BC.

a As the Radius Caer 90 00 ©O
ie To the Sine of the Side given 29-53" 43
% So the Tangent of the oppofh ite Angle 23 3% 30

To she Tangent of the Side reqnired UY 39 43

next the Side reqwired. -

BD C elas. 39 min. CO And DG,
At he Tangent of the Argl e 38 30 00

j To the Tangent of the Sidegiven 11 30 47
$0 the Radius 90 oo 00

To the Sine of the Side required 14 50 It

ae - the Side required.
af
rf 3 Zgr. 31m,

AS the Radivs

Tothe Cofine of rhe Angle | 23 31 «30
- Sothe Tangent of the Bafe ge Weanoo

30 fee. to find the Side A C.

To the Tangent i the Side. hana 27 53 43

AAA 2

. ss % To fd A Side, by knowing the other Side, ie the

g0. 00 '60

o -As in the Reratot ACB, having A C 27 gr. 5 3 vishes 43 fe and

10.0000066 »

9.6901 I12 |

-9.6388199

ee ey

19,308.93

10. To find a Side by, knowing the ihe Side, and the Angle

> Asin the Rectangle BC D, vivine BC r1igr. 30min. 43 fers ose

9 obceug?
93089311
T0,.0000000

9.408359

TI, T he a ‘Side by knowing the Bafe and the eAagle next

Rs ‘athe ReGtangle ACB, having A B 30 gr. 00 min. and B AC

10,0009000_

9 9623153
97914393
19 7237546
32, T0

7

5 a Se a ce

A al The general Vest the Camm i:

12. To find the Bafe by) knowing both the Ob igu sae

_ Asin the Reétangle ACB, having B AC 23. £¢. 31 min, 50fen 4
ABC 69 gr. 20m, 35 fee. co find the Bale AB. |

eds the Tangent of the one Angle 23 3K 30 9.6388 19g) 4

c To the Cotangent of the other ‘ 69 20 35 9.5763 sos. |

So the Radins ’- 9° ©0O 60 ae
To the Cofine of the Bafe 30 00 ‘eTo) 9. 9375306

13. To find the Bafe by knowing one of the Sides ad the
eingle next that Side.

PAs i in the Re&tanole AC B, having AC 27 ars

5 : min. 43 fee. ahd it |
B AC 23 £7+ 31 min, 30 fee. to find the Bate AB |

As the ¢ ofsne of the Angle 23, 31 30 9.962318 4

. 1s te the Radius 990 ©6060 66 10 Cec00ca
So the Tangent of the Side 27 +5343 9.72375 47
70 the Tangent of the Bafe 30 00 eo 9.76 14394.

4. 70 find an Angle by knowing both the Sides.

Asin the Re&tangle A C B, having AC 27 or.

3 min. 43 fee, and
’ BC u1g¢r. 30 min, 43 fee. to find the Angle A B C, |

ets the Sine of the next Side II 30 43 9.300052 ": |

Is to the Radius © ©0 0©O 46, ©CO0C00 | oh
Sethe TangenPof the oppofite Side 277-53 439.92 37547

; : a ]

Lothe Tangent of the Angle 69 20 35 10,41 3649 5 |

15, To ‘j |

|

\
an

‘Cries 2h Gack” b PDT Vane ele "ote a 4 all we
bee Ss Y Creegee Les hie Les | ae a Pa OTA Fag |S ete ye PS »’ 7

fk Dh eA = fae awe ah hart Puy " Mi 3, vane y P
7 y) il >

15, To find an Angle by knowing the Bafe, and the Side next
: the eAzgle required.

Sand Tables of Logarithms, 189

| ee
Asin the ReGtanele BCD, having BD 18 gr. 41 m, 56. /eee and

BC u1gr. 30m. 43 foc. tofind the Avgle B DC.

As tke Tangent of the Bafe 18° qi 56 9.295063.
_ Tothe Tangent of the Side IE 30 43 9.3089311
So 1s the Radius eat 90 ©8 09 10,0000000'

To the Cofine of the Angle 53 00 46 = 9.7794248
16, Tofind an Angle by knowing the Bafe and she other Oblique Angle.

_Asin the ReGtangle A C B, having the Bafe A B 304r. and BAC

> 23¢r. 31m. 30/fec. to find the Angle BAC. |
As the-Cofine of the Bafe — 30 00 00 9.9370000.

Is to the Radius g2 90 ©O ~~ 19.0000000
So the Cotangent of BF wngle given 23 3% 30 103601801

a @ the Tangent of the Angle required 69. 20: 35 = 10.42 36495

Thefe 16 Cafes are all that can fall our in a Rectangle Triangle. Thole.
which follow do hold in any Spherical Triangle whatfoever.

Inany SPHERICAL TRIANGLE whatfoever.

17. To find a Side oppofite t0 an Angle given, by knowing one Side and two

Angles, the one oppofite tothe Side given, the other tothe Sid: required.
Asin the Triangle ABD, having AB jogr. BD C 38 gr. 30 m

andBA D 23¢r. 31m. 30 fee. tofind the Side BD, whieh here re-

prefenteth the Amplitude.

t

Asthe Sine of the next Angle 38 30 00 9.7941495-
Tothe Sine of vs oppofite Side 30 00 00- 9.6989700

| : “ee 9ST TOS:
Sothe Sine of the oppsjite Angle 22h. FL» 30, 9.601135

Fo the Sine of the Side require 4B 41 «56 9: 5052 05%
hie” «ge ‘

ER se 4 The general Vfe of the Canon, tah)
"Or changing the Site of che two middle Terms. . = ie

et Complement of the Laticude, P $ the Com-

ZS 40gr. to findthe Angle Z PS, fubtending the Bafe ZS,

NS DF Gi dN ab ia wih coy
aire at

As the Sine of the next Angle 39 30° ‘06: ; 9:7 941495 bE
Yo the Sine of the oppofite Angle 23 ae ae 9.6011353
ny pie ie a : be 1930 143 7 A es
Sothe Sine of the Side given 30° 00. 00 9.608 oTeE S989700 Ml

To the Sine of the Side required 18 41 56 965059557 a
And fo writing this Difference 1930143 ina Paper by icfelf, and
holding it co che Sine of che Sidein che Canon I gr. 2,354, §, and {o im
forward, ic would be no long work to fubtraét, and write the Remain- y
der ina Column by icfelf, and fo find the Amplitude for cach Degree
aud Minute of the Ecliptick. aa | ...
Or inftead of {ubtracting this Difference, we might firft take the fame ba
out of the Radius, and then add the Complement as I fhewed before, in @
the general explication of the Rule of Three. a

18. To find an Angle oppofite to a Side given, by knowing one Aagle and two % ;
Sides, the one oppoftte to the Angle given, the other to the Anglerequired,

As in the Triangle ZPS reprefenting the i Zz
Zenith, Pole, and Sun: where Z P is the ‘,

plement of the Declination, ZS the Com-
plement of the Sans Alticude, P ZS the
Azimath, Z PS thehour cf the day from
the Meridian, and PS Z the Angle of the -
Suns Pofition in regard of the Pole and Ze« 5
nith; having PZS, 13097. 3 mis. 11 [ee

PS 7ogr. and ZS 40 gr. to find the Angle ZPS,

As the Sine of the next Side 70 60 ©0 9.97298 8 i |
Ts to the Sine of hes oppofite Angle 130 03 1. 988590 a
ee , 890705
So the Sine of the oppofite Side 49 00 00 69. 808067<. |
To the Sine of the Angle required ~ 3K 34 26 : See i
e 4 |

‘eo

19. To find an Angle by knowing the three Sides.

-, 4
As in the Triangle ZP S, having ZP 38 er. 30min. PS

.
7Ogreand
i 4

| eta and Tables of Logarithms. == 19K poe:
As the Reftang'e conteined under the Sines of the Sides, #5 to the Sqnare
of the Radius: ; ae | “
|” So the ReLangle conteined ander the Sines cf the Half-Sum of the three
» Sides, and the Difference between this Half Sum and the Bafey — ~
—- Tothe Square cf the Cofine of half the Ankle required.
The Bafe fubtended is 40 gr. OO me.
2. The two Sides including the Angle i ie
The Sum of the three Sides M148 Le 30%
The Half-Sum of thefe three at te iy
The Difference between this and the Bafe S44 hee a
| Here for the Square of Radius we take 20,0000000, to this weadd
9 9833805 the Sine of 34 gr. 15 min, and 9.753579 che Sine of 34 gre
15 min. which make 39.7337384. :
| "Then for che Re€tanele of che Sides, we add 9.7941495 the Sine of
38.gr. 30 win. and 9,9729858, the Sine of 70 gr. which make.
— 19.7671353. This we take out of 39 7337384 and there remains for
the Logarichm of the Square 19.9666031, the half thereof 9.9833015 |
~ we find ro be the Cofine of 15 gr. 47 min. 13 fee. And fo the whole
| Angle required is 31.g7. 34 min. 26 fec. :
| Or for fuch Numbers as areto-be fubtra@ted, .we may take them out
» of the Radius, and write down their Complements, and then add them
~ together wich the reft, the manner of the work in either way willbe fuch.
| as followeth...
40 gr. OO m3,
38° 30 9.79414.9§ 2058505"
70. 00. 9.9729858 _ 270142
| ek 30 19 7671353 :
pat» ne 9 9831805 9:98 33805.
34: 3S. 9: 7§93989 | 97593579
| 20,0000000"
iP 39.7337384 | |
iy 19.9666031 Lhe {Bho fee 19.9666031
9.9833015 1G. 4B 33 9.9333015 »
31 34 26 .
Tn

= i> te ee, ee) es Pw ODS Bares ons tron t, Be Se” 9! ‘ ae
ia acelin er a ale ue is Maan re Sacco seh E Ws ia aie: ih ibys DS oo Sie
’ : aes Pie ne F 5 ae Si ines at i

hays

igz(. The general Ufe of the Cann =

Inthe like maaner we may find che Angle PZS tobe 13 ofr. 3m.

An fes. and che Angle ZSP 30gr. 28 m. 11 fee, --
Naga Ty find a Side by knowing tbe three Angles.

If for either of che Anglesnext the Side required, we take the Com- _

plement to 180 gr. thefe Angles willbe turned into Sides, and the Sides -
into Angles. Then may the work be the fame as in che former Propofition. ~—

_ Asin the Triangle Z PS, knowing the AneleZ PStobe 3 gr. 34m,
26 fec. PZS 130gr. 3m. 11 fec. and ZSP 30 gr 28m, 11 fec. if
it were required to find the Side Z S$ oppofite to the Angle Z P Ss, 1

would take 130 gr. 3m. 11 fee. out of 180gr. the Remainder will
be 49 . 56 “ 49, : ,
Then, asif I had a Triangle of three known Sides, one of 31 &r. 34m.

26fec. another of 30 gr. 28 m. 11 fec. and the third of 49 gr. 56 my, dt
49 fec. Iwould {eek the Angle oppofite to the firft of thefe Sides bythe

laft Propofition. | - |
So che Angle which is thus found would be the Side which is here re- |
quired, | | | ao |
Thushere the Angle oppofircis 31 34 26
The leffer of the next Angles _ 30 28 or 9.7050799 ] |
The Complement of the other 49 56 40 9.8839153
The Sum of thefe three Ii 59 26 : 4
~The Half Sum ; 55 59 43. 99185490
The Diff. from theopp, Angle 25 25 17 9.6164170
The Sum of double the Radius and 20.0000800
The Sines of Half Sum and Difference is | ‘39.5349660_ 4
Take hence the Sines of the next Angles 195889943
There remains for che Square | ‘19.9459717- |
. The half whereof is sieves 58 |

the Cofine of 20 gr. 00 ms. and {o the Side required, 4Ogr. 00 m.

The other Sides may be found in the fame fort; but when we know
-eicher three Sides and one Angle, or three Angles and one Side, thereft .|

may be found more readily by the 37 or 18 -Propofition.

2t, To

. ana Tables of Logarithoss. ? AOI?

21, To find a Side by having rhe other two Sides and the
Angle comprehended. |

‘

——

_ the oblique-angle Triangles given, into two Re&angles.

Ps

| Asin the Triangle Z PS, having Z P 38 gr. 30 ™. PS 70 gr.00 m.

and ZPS 31gr. 34m. 26 fec. to find the Side ZS. |

~ In that we have ZP and ZP S$, we may fuppofe a Perpendicular
~ ZR to be let down from the Angle at Z upon the greater Side P S:

_. Triangle ; if more than gogr. it will fall without the Triangle, upon
_. the Side produced, and divide the Triangle given into two Rectangles
ie Z RS andZ RP. Wherein |

r, We may find the quantity of this Perpendicular by the firft Pro-

& pofition of Spherical Triangles-
| 2, Wemay find the Side P R either by che fecond or tenth, or ra-

ther by the eleventh Propofition : w

wa)

Io RS. oo
3. Having ZR and RS, we may Gnd the Bafe ZS, by the fourth

_. Propofition, as I thew in the ufe of the Sector.

i
|

| But here for variety Iwill thew how the fame may be done at two
_ Operations, both in this and the reft of the Cafes following, witheut

) Knowing che quantity of the Perpendicular.

This and the Proportion following are beft refolved by reducing

So if Z PS the Angle given be lefs chan gogr. it will fall within the |

hich Side PR will give the Side-

. ae Bbbb | As

a fy? a Ve oy
La EON RTN SEN eee iden Thad ih ga) Se ert > pC ED a
a yi Tera wy , Y ree lava At! ase py Sot ES Buse i: sara) Me Eeae sa as does

. a n

0194) The general Ufe of the Canon 6,
(1s Asthe Radinn or Sine of ZRP 99 08 09 10 cPCCeC0
. ; ; * ee Cee
To the Cofine of the Angle ZPR. 33 34 26 9:9394233
So the Tangent of the Side ZP 38 30 oo 909006052

To the Tangent of the Ark PR 34 - oF cae) 19,83 see | |

2. Asthe Cofine of PR On 67.5 50 99179342 |

To the Cofine of Zip 38 3@ Co 9-893 5443

his: or | / : 243899
To the Cofine of ZS

42 ©0 oo 9.8842 539

22. To find a Side by knowing the other two Sides and one |
“ingle next the Side required, |

Asin the Triangle ZPS, having ZP, 38 gr. 30m, hae ZS 40 gr.
60m. and ZPS, 31 gr. 34m. 26 (ec. to find the Side PS, f
¥, Find the Ark PRb

y the eleventh Propofition as before.
2. Asthe Caofne of P

38 30 00 = _9.893sa4y

To the Cofine of PR 34 97: 39 _9.9179342
ee ne

: 243899

So the Cofine of ZS - 42 ©0 00 9.884 2539
Lothe Cofine of SR A

3$ 52 30 © 99086438

123,98. 7% find a Side by knowing one Side asa the
wext the fecond Side,

As in the Triangle Z PS, having ZP 38 er. 30m. ZPS 31 gr. |
34m, 26 fec, andZ§P 39 gr. 28m. 1 I fec, to find the Side PS. |

1. Find the Ark PR as before,
2. Ms the Tangent of ZSP

two Angles

f 30 28 qr 97696336
othe Tangent of ZRS 3 34 26 = -9.7885746 |

pal 189510

So she Sine of PR 34 07 30 9.7489617
To the Sine of SR 33 $2 30 9.76798 +4

24. To

Cae Nee “OMPS e Le a Ns os ee dee ce Oe Pe ea r ee *. _<
ere WS feet hi a WYP ats a ROPE. PLAN ny nly $ ~) Passe oe . YAS Dee A ae
a a | aan bans ; z A iy =

.
ie

a if and Tables of Logarithms. eae 19¢

a4. To find a Side by knowing two Angles and the Side
|e : inclofed by them. — eae
_Asinthe Triangle ZP S, having Z P 38 gr. 30m. ZPS 34 gre 34m:

26 fec. and PZS 130gr. 3m, Ut fee. to find che Side Z S.

LAs the Cofine of - PZ 38 30 08 9.8935443
| Is tothe Radius - : 99 ©O 00 10.000C000_
\ So the Cotangent of LPS 31 34 26 1O.21142§ 3
i | eee
| To the Tangent of .PZR 64 18 50 10.3178810
a As the Cofine of . SZR 65 44 22 9.6137228
To the Cofine of PZR 64 118 50 9.636935
= 232083
‘So the Tangent of PZ 38 30 SO 9.900605 2
| Torhe Tangent of ZS “Fe AOm OO. GO 9-9238135
| 25. To find an Angle by knowing the other two Angles

and the Side inclofed by them.

As in the Triangle ZPS having ZP 38gr. 30m ZP § 30 gre
34m. 26fec, and PZ $130 gr. 3m. IL fec. C0 find the Angle ZS P.
1, Find the Angle PZR by the fixteenth Propofition as before.

(a, As the Sine of PZR 64 18 50 9.9543122
| To the Sine of SZR 65 44 21 9-9593453
| * 5O33E
| So the Cofine of ZPS 32.34 26909304223
To the Cofine of EP zo 28 It 9.9354554

hs 26. To find an Angle by knowing the other two Angles and
| one Side next the Angle required,

oo -
i

) UASin the Triangle Z PS, having Z P 38 er. 30m, ZPS3t gr. 34
26 fec; and ZS P 30gr. 38m trfee. tofind the Angle PZS.
| ot Bbbb 2 t, Find.

ee te!
i ae cae
Pe ee ee SnD BS =

TAN DUEL Me OEE gt erate read) GRAS CM Ee SR Ee a
Z ae y bint - Wh OAT mb ae \
‘ od wa v-

196) | The general Ufe of the Canom —
4. Find the AngleP ZR asbefore.
2. As the Cofine LPS). 3% 34 26 99304223 ue
To the Cofneof ZSP 3D 2B at 99354554
eat § ) 50338 e
So the Sine of PZR 64 18 0 9.9548122 :
To the Sine of - TSI AR

1 64 44 21 9.959845

Gal
2.

27. To find an Angle by knowing two Sides and the
= taesivan Angles contained by them, ~

As in the Triangle Z P S, having LP 38¢r.. 30m, PS7o¢r, and |
ZPS 31 gr. 34m. 26/fcc..t0 find the Angle ZS P, He |

4h

T, Find the Ark PR as before.
24 As the Sine of SR 3h ‘ig2. 30

i
|

9.76791 29

Tothe Sine of PRo: 34°97 3 967489617

. | 189510 ©

So the Tangent of ZPS 32 34 26 9.78857 .46 :
To the Tangent of ZSPR vviZOP 2B 4iuz 9.7696236

28. 70 find an eAlngle by knowing two nest Sides , and |
| one of the other gAngles, AC Sr. |

Asin the Triangle Z PS having Z P 38 Lr. 30m. ZS
ZPS 31 gr. 34m. 26 fec.to find the AnglePZS.
1. Find the Angle PZR as before,

40 gr.. and

- 2. As the Tangent of Z§ 40 00 OO 9.9233 81 35
To the Tangent. of ZP 38. 03 oo 9-900605 2
| =o" "23908 |
Se the Coffne of PZRs..:64 18459 .»9:6360g0m
70 the Cofixe of SZR 6544.21. 9,6137228

Thefe /

|

26, and 28. Propofitions may

\f

=

See ae and Tables of Logarithms, = = 97
Thefe 28 Cafes are thofe which I {et down in the ule of the Sector,”
and all chat are commonly required in a Spherical Triangle. I will here

add twomore, to fhew how chat which is found before by the 22, 23;

fomecimes be found more eafily, wiz

29 To find a Sides by knowing the atber two Sides, and
se their oppafite Angles.

As inthe Triangle ZP S, having PS 70 gr. and PZ S 130 ¢7. 3 M-
it fec. together with ZS 40 gr. and Z PS 31 gr. 34%. 26 fee. (0
find the third Side Z P. ad

As the Sine of balf the Difference of the Angles given,
To the Sine of Yalf the Sum of thofe Angles:

So the Tangent of balf the Difference of the Sides givens ,
To the Tangent of half the Side required.

30, To find an Angle by knowing the other two Angles, .
and their oppofite Sides.

Asia the Triangle Z P.S, having the former parts PS, P ZS, LS»,

and ZP 5, to knd the chird Angle ZS P. | Ra

As. the Sine of half the Difference of the Sides given,
To the Sine of half the Sum of thofe Sides : |
Sathe Tangent of half the Difference of the Angles given,..
To the Cotangent of half the Angle required,

198 rhe general Ufe of the Canon

CHAP. Ill.

Concerning the joynt Ufe of the Linas of Numbers, Sines
ana Tangents. |

Oncerning the joynt Wife of the Lines of Numbers, Sines and
Fangénts, I thewed how they might ferve for the Refolution
G of Right-lined Triangles, whereof I fec down five Propofitions
in the ninech-Chapter of the firft Book of the Crof-faff, And thefe
alfo may be applied co the Table and Canon of Logarithms, ~ .

The Sides of thefe Triangles are meafured by abfolute Numbers,
and fo reprefented by Logarichms. | eae

The Angles are meafured. by degrees and minutes, and foto befound —
. by Sines and Tangentsin the Canon.

PROP. 1.
Having three eAAngles and one Side, to find the other two Sides.

ie it be a Rectangle Triangle, wherein one Side about the right Angle
being known, it were required only to find the other, thismight be 7
_ readily done by Sinesand Tangénts. As in the Rectangle AI B, know-

ing the Angle BAT to be 43 ¢r. 20m. and the Side AT tobe 244, if
it were required to find the other Side A I. |

J

“4s the Radius (the Tangent of) 45 gr. OO m.

10,0000000
Is tothe Tangent of the Angle 43 30 9. 9749195 |
So # the Sidegiven —- AI 244. 228 2.3 $9 3898
To the Side req sired B I 230 202 22,3621093

But .where both the other Sides are required, itis beft done by Loga-
rithms and Sines. Asin the fame Reétangle A I B, having the three
Angles and the Side AJ, to find both BI and AB.

i

pu ieys

‘ag « avd Tables of Logarithus, ; a9 |
As the Sine of the oppofite Angle ABI 46 40 9,8617575
Isto the Side given’ Al 244) 900 2.38735 98

74743675

I}ie
| -

_—————

| So the Sine of the fecond Angle BAT 43. 20 9.836 477°

To his oppofite Side Bt 34 APS 2.362109 3
| And the Sine of the third Angle AIB = 90 ©09 = ~10,0000900
To hes oppofite Side AB 35.5007 2.5 2563 23

The like holdech alfo in Oblique-angled Triangles.

| As inthe Triangle A BD (which I propofed formerly as an example
for the finding the Diftances) where knowing the Diftance between A
| and D, tobe 100 paces the Angle B A C to be 43 gr. 20m. the Angle
_ BDA 123, orthe outward Angle BDC 58 gr. and confequently the
_ Angle AB D'oppofite to A D the Side given, tobe 140 gr. 40m.
it was required to find the Diftances A Band DB. |
As the Sine of the oppofite Angle A BD 14 40. 94034554

| Is to the Side given AD 100 ~2%2 = 2,0000000
| La pues 74934554.
| Sothe Sine of the fecond Angle ADB 53 00 )=—- 9. 9284204
; To hes oppofite Side | AB Ye ey 52496 50
And the Sine of thethird Angle DAB 43 26 9.8364770

| To hes oppofite Side DB 271 *42 3.43 30216

ly,
PROP. If.

| Having two Sides and one Angle oppo/ite to either of thofe Sides, tofind the

| ather two Angles and the third Side.

le

| ee -Sinthe Triangle ABD, having the two Sides A B 335. paces ;

« fA and AD toopaces, and knowing the Angle ADB which is

| oppoficeto the Side AB, to be 122 gr. or the outward Angle B DC to

{be §8gr. if it were required ta find the other two Angles at A and B,

and che third Side BD, I may firft find an Angle ABD oppofite to

> the other known Side AD. | i
: i

200 The gentral U[e of the Canon “-
As the oppofite Side = * AB) °335 *** - 2g250448m
Tothe Sine of the Angle given ADB 58 00 9.9284204
74033756

Sot the next Side A AD 100 222 = 2,0000000 \

To the Sine of hes oppofite Angle ABD 14 §9%' 9.4033756

Then knowing thefe two Angles atc D and B, JI take the inward
Angle ABD 14 gr. 59m. s0/fec. out of the outward Angle BDC
S8gr. 00m. and {o find the third Angle BAD, tobe 43 gr. 20 mm;
- rofec. So having three Angles and two Sides I may well find the third.

Side B D by the former proportion. |

As the Sine of the firfé Angle ADB 58 00 9.9284204.

1s to hie oppofite Side AB 335 =** = 245250448
74933756

So the Sine of the laf? Angle DAB 43 206 9.836503.

To his oppofite Side DB 27ts 322 2.4331 299)

PROP.

”

===

oe } ba
| ff “¢

~ and Tables of Logarithms. . 20%

PROP. Ill. |

Halog two Sides, and ehe Angle between them, to find the ether two
“a Angles and the third Side. — :

'E the Angle conteined between the two Sides given bea right Angle,

the other two Angles will be found readily by Tangents and Loga-

‘rithms. Asin the Rectangle AIB having the Side Al 244, and the

Side 1 B, to find the Anglesat A and B.

As the greater Side Al 244 2. 3873898
Is to the leffer Side IB 230, 2, 3619278
So the Radivs, the Tangent of 45 gr. 00 ms. 10. 0006000
Tothe Tangent of theleffer Angle 43 18 5 9.9743380

~ Bucif ic bean oblique Angle tharis conteined between the two Sides

- given, che Triangle may be reduced into two Rectangle Triangles, and
_ then refolved as before.

Asin the Triangle A D B, having the Sides A B 335, AC 100, and
the Angle B AD 43 20’, to find the Angles at B and D, and thethird

SideBD. Firft, 1 would fuppofe a Perpendicular D H to be let down
from D, the end of che leffer Side, upon the greater Side AB: fo thall
-Thavetwo Rectangled Triangles DH A and DHB. And in the Re-
Gangle AH D, the Angle at A being 43- 20, the other Angle ADH
will be 46 40° by Complement, and with thefe Angles and the Side

AD, I may find both AH and DH by the firft Proportion. Then ta-
king A Hout of AB, there remains H B for the Side of the Retangle
DHB, and therefore with this Side H B and the orher DH, I may

~ find the Angle ac B, by the former pare of thisProporcion. And wich
this Angle and the Perpendicular D H, I may find che third Side DB,

by the firft Propofition, .
Or having two Sides and the Anele between them, we may find the

other two Angics without letting down any Perpendicular, in this

omanner.

As the Sam of the two Sides given,
Is tothe Difference of thefe Sides: |
So the Tangent of balf the Sum of the two oppofie Angles,
Lothe Tangent of half of the tise between thofe Anglese
Tea So

— >

>

no The general, Ufe of thé Canon |

ia. ANY ae Oe ee ee, ‘x s
» ot Che eh sh pA TANTS SRS AL > em Sh mtn cot Wl CIE DS Weak es So tehe
A CE ae as ae So TP seh athe deg ;

fae “ fag f

D ;
So here having the Side AB 335
and the other Side aa VAD os an Oi

The Sum of thefe Sides is 435
and the difference of thefe Sides 225
The Angle conteined B A D js 43 20
The Sum of therwo Oppofite Angles 136 40
The Half Sum of thefe Angles 68 20
~ And by Proportion and half difference 53 40 4
This half Sum and half Difference make 12 204 the
and the Difference between them 14 ¥9 3 the lefler Angle,

PROP. Iv.
Having three Sides, tg find the three Angles,

; Hi one of the three Sides piven be the Bale (bur rather the greater Side)
thac the Perpendicular may fall within the Triangle. Then gather —

che Sum and the Difference of the two Sides,and the Proportion will hold,
As the Bale of the Triangle, ; |

To the Sum of the Sides -
So the Difference of the Sides,
To the alternate Bafe,

ee a fee ae te

vee

a Perpendicular from the oppofite Angle, it fhall fall upon the middle of
he Remainder. Asin the Triangle ADB. . - be

‘ment A H, the Diftance between the Angle at A, and the Perpendicular

- TrianglesD H Aand DHB, inwhich having two Sides, and the righe
Angle, we may find the other Angles by the fecond Proportion.

_ Angles in all right-lined Triangles.
. PROD. V.

Having the Bafe and Perpendicular in a right-lined Triangle, to find the
| | faperficial Content. Haines is
q “He Perpendicular may be found by one or other of the former Propo-
$B ficions, and-that being knowa we may find the fuperficial Content.

dicular DH 68545. te
©.3010700

As the Namber of 2 |

© To the Perpendicular 63.545 tn acladg MSR OAT IT
Se | | 1.5349457
Bo the Bafe- 335 wet 2.5250448
© To the Content . 11481 a 4.0599905

| Orif we would find the Contenc without knowing the Perpendicular,
peanay put two or more Operations into One, “as in the Proportion fol.

4

P and Tables of Logarithms. (202.
| This alternate Bafe being taken forth of chetrue Bafe, if we let down

| The leffer Side AD 100
| The other Side es 2) 270 ‘
The Bafe of the Triangle AB 335 2.5250448 .
— TheSum of the Sides 371 2. 5693739 ~
: ; ioe 443290 ~
The Difference between thefe Sides 171 2.2329961
And fo the alternate Bafe is 189 74% 262773252
This taken out of 335 leaves 145, 945
The half whereof 1s 7 72 812. And fuch is the Seg-

DH. $Sothat having drawn this Perpendicular, we have two Rectangle —

Thefe four Propofitions may fuffice for the Refolution of the Sides and

“Asin the Triangle ADB, having the Bafe AB 335, and the Perpen-

Cece 2 etna DRG

FE aA et ee She Ys ee SE bas 3 Ore ;
‘> a5 : ;

204 | The general Ufe of the Canm
Bain : PROP. VI.

Having two Sides of a right-lined T riangle, and the Angle between
them, to find the Content.

Dd the Sine of che Angle, and tae Logarithms of both the Sides,
A A from the Sum of thefe-fubcract —10-3010300, fothe Remainder
fhall be the Logarithm of the Contenr,

Asin the Triangle A DB, having the Sides A B 335, AD foo,
andthe Angle B A D 43 gr. 20»,

_ TheSine of the Angle 43 gr. 20m. is 9.8364770
The Logarithm of the Side PaNG Shee 4 aay 265250448
The Logarithm of the Side AD 100 2,0000000
The Sum of thefe make 14,3615 218
From which fubrraé the folemn Logarithm
The Remainder will be |

The Logarithm of 3 1494 the Content required,

PROP, VII.
and one Side of a right-lined Triangle , to find

10.39010300_
4.0604918

|
Having three Angles, |
the Content.

|
Dd the double of the Logarithm of the Side given, and the Sines of |

tie two next Angles; from the Sunt of thefe fuberad the Sum of
¥0.3010300, and rhe Sine of the oppofite Angle, fo the Remainder —
fhall be the Logarithm of the Content. : |
Asin the Triangle A DB fappofing the AnglesB A Ctobe

a ||
BI]

|

a |

34,8 20m,

BDA 122 gr,com. ABD 14 ¢r.4Om.and che Side A Dro the 100 parts...
The Logarithm of the Side A C. 160 is 2.0000000
‘The fame again | 2,0000000° |

_ TheSine of the Angle BAC 43gr. 20 m, 9.8 364970 —
The Sine of the Angle BDA 58 0 9 9284204. :
The Sum of chefe four make 23.7648974
#igain, if weadd the folemn Logarithm £0.3010390
To the Sine of the oppofite Angle 14 gr. 40 m, 9-4934554 |
~The Sum of both will make , 19.70448 54 |
Which fubtra&ted frem 2 3-7648094 leave 4,0604120 |

The Logarithm of £ 14.92.che Content required,
. PROP,

Differences. That done,

* Having the rhree Si

‘and Tables of Ligarithmss 285
PROP. Vill. —

"Having the phyee Sides of avight-lined Triangle, to find the Content,

Irft, fet down the three Sides, the Sum of them, and che Half- Sum.
Then from this Half-Sum fuberact each Side feverally and note the
add the Logarithms of che Half-Sum, and thefe.

Differences, the half thereof thall be the Logarichm of the Content.

Thus in the Triangle AB 335

AD B, the. three DB As 4247

Sides are AD 199
The Sum of thefe Sides 1s 906
The Half-Sum | 352 3.9477747°
The Difference from AB 18 1.25 oe }
The Difference from DB $2 119138138
The Difference from AD 253 2.4.0 31205
The Sum of their Logarithms | 8.1199815

4.2599907:

And the half thereof is
The Logarithm of 11481

PROP. Ix.

223 the Content required.

des of aright-lined T: viangle, to find the Perpendicnlars

Sin the former Triangle A DB, to find the Perpendicular 'D sat
PX Firht, find che Content of the Triangle by the former Proportions...
chen may the Perpendicular be found by che converte of the fifth Propo-

f{rion.
335 2.§250448

As the Bafeof the Triangle
To the Superficial C ontent 1148444 410599907?
15 349459:
| So always the Nursber of 2 0.3010300
To she Perpendicular 63,244 : 1,8359759

PROP.’

Heh PP eal Re ae rk Pe San Toor ee Sesh : Bi eam nn doe fh NE an zs Wee, a pats :
ay oe : ' in mt ct :
‘ ; ,

206 | | The vineral Ofe of the Canes

PROP, X.

— Having the Semidiameter. of a Circle, to find Mei bet, fae sans vk i

prepofed.

S if in protracting the former Triangle A DB, ic were required to
find the length of a Chord of 43 gr. 20m. agreeing to the Semi-
diameter Ab, ‘which we fuppofe co be three inches. . This might be done

_ by the firft Proportion, for if the Chord were drawn from E toF we

fhould have a Triangle E A F of three Angles and two Sides known. But, ~
more generally comparing the Sine of 30 gr. with the Sine of half the
Ark propofed, the Proportion will hold.

“ As the Sine of the Semi-radists 30 gr, 00m. 9.6989700
hy the Semidiameter gee 0.4791212
| 9.2218488
So the Sine of half the Ark ~ at gr. 40m, 9.5672689
To the Chord requived 2 24S 0.3454207

So that having drawn the Line AE, and defcribed an occult Ark of a

Circle upon the Center A, and Semidiameter AB, at the Diftance of |
_ three inches, if wetake out two inches, and 215 partsof 1600, and ine

fcribe them into that Ark from E to F, the line A F thall make the Angle
FAEtobe 43 gr, 20m. as was required. a

Thus having applied that to the Canon and Tables of Logarithms
which Thad fet down before for the general Life of the Lines of Numbers,
Sinesand Tangents, ic may appear fufficiently, that, if we obferve the
Rules of Proportion fet forth by others, aud work by thefe Tables, we
may ufe Addition inftead of their Multiplication, and Subtraction in-
{tead of their Divifion, and fo apply thefe general Rules to infinite par-
ticulars. i

CHAP.

-~

Having the Side of a Regular Fort, with the length of the Gorge, the Flank:

| The Face

tity of the Angles‘belonging to each Fort, beginning withthe Quadratee

eee Gre ome oe CRIT te Po CAM cn fy Ie ot gear Ma SON SoG ga
4 > i Mi i + ‘5 a

Mc CCHAPLLY,

Conteining fome Ufe of right-lined Triangles in the Prattice

of FORTIFICATION.

1, That the Angle of the Bulwark may be eithera right Angle or

; N the late manner of Fortification the ordinary Care is :

near Unto ite “ . :

2. Thacthis Angle may be defended from the Flank and Cortin on
either Side. | ’ .

3. That the Linegof Defence may not exceed the reach of a Musket,
which is faid co be twelve fcore Yards, and thofe make 720 foot.

4. That che depth of the Flanksand the bredth of che Rampart be fuf-
ficient co reGift battery ; and chat may be about 100 foot at the ground..

Upon thefe confiderations depend the reft of Lines and Angles : where-
of I will {et down fome Propofitions, beginning with that. which may-:

-refolve the works of others...

ve

PROP. I:

and the Face of the Bulwark, to find the reft of the Lines and Angles. —

each Bulwark like unto other. 3
Suppofe,chac by obfervation or otherwife,we have found that in a fquare

Fort, the Side was 7e0 foot, the Gorge 140, the Flank 100, and the Face
335.: Ina Pentagonal, Hexagonal, Heptagonal, as inthis Table, -

———
_— OO Se

———

er

The Side AB | 700 | 800 | goo | 950 |1000
The Gorge. AD-|140 |-180 | T90 || 200: | ‘230
The Flank PD Bt o0 Pad fiat go Prgb dirs 8s

__EF ! 335 | 352 | 370 | 360°| 420°

-And thac ic were required: to find chereft of the Lines, and che quan

Fir,

aud Tables of Logarithms: - eg 8

Regular Fort isthar, whichis made with equal Sides and Angles, .

\Quadr, |Pentag. | Hexag \Heprag. OcFagon

Firft, we may prorrad this Fort, by making a Square whofe Side A B

fhall be 7eo foot by the Scale : then take but 14° for the Gorge, and fey

them off from A unto D, and from A unto H: at D and H raife two

Flanks perpendicular to theSides of the Fort, and there prick down 100 |
from D unto E, and from H unto G. Thae done, take 335 out of the —
{ame Scale, and fetting one foot of the Compaffes in the point E, makean_
occule Ark ofa Circle. Again, fetting one foot of the Compaffes in the —
point G, make another occult Ark, croffing the former in the point Fs |

So che Lines EF, EG hall reprefent the Face of the Bulwark,

In like manner for the Bulwark at B, we may fet off the Gorge from |
B unto N, @c. So have we divers Triangles, which may berefolved by
the firft three Propofitions of right-lined Triangles: and the manner of |
it. fhall be fo fer down, asthat Precept may be eafily diftinguithed from —
the Example) and applied to any other, not only by this Canon and —

Table of Logarichms, but by the old Canon of Sines and Tangents, and

by the Lines of Sines and Tangents both upon the Sector and the

Crols Raff. j

1. Imthe Re&tangle A DB, having the Sides AD, AE, we may
find the Anglesac A and E, and the third Side AE, -by the former part”

of the third Proportion of Right-lined Triangles.

|

a

|

|
i

-F, by the third Propoficion of right-lined Triangles.

ae 4 Oe ie ee MTN ee AN See nL te he Bil bo vim. wh a . a
tt aie s af Abe iia a “Ee ca rory or = Gis a ; oil
{Piva A

| whales of garit. 89

As the Gorge AD 140 cA: 2.1461280
f To theabitank DE 106. “2,.6000000.
Sethe Radius — Ys 90 00m. 20/ec, 10.0000000

To the Tangent of DAE 35°322 - 9.85 38720

Take the Angle DAE out of gogr. the Complement will give the Angle |

DEA; andthen,having two Sides and chree Angles,we may welLfind the
third Side A E by the firft Propoficion of the right- lined Triangles.

As the Sine of DAE 35 324 9.7643542

- To the Side DE~-~. 100 “ 2,0900000
So the Sine of ADE _ 90 00m. oofec. 10.00000CO,

_ To the Side AE WET hasan 2.2356458

2. Becaufe the Fort is fuppofed to be fure, the Angle HAD, mutt be

— gogr. andthe half AngleC A D 45 gr. if we add this AngleC A D unto”
~ “the Angle D A E and take the Sum our of 189 gr. che Remainder 99,27 2

thall be the Angle E A F. Then in the Triangle E A F, having the Angle
at. A, and thecwo Sides FE, A E, we may find the other Angles at Ean

ids the Face EF 235-i teil 9 2.5 250448
_=To the Sine of EAF . 99 32S ade haatyit 9:9940502.

| 724690054

So the Line ALE Ge thee. 2.2356459
Tothe Sine of AFE 30 26% 9.7046513

: 5 .
Addthis Angle A FEto,the Angle E AF, and take the Sum outsof

5 180 gr.the Remainder 50 gre 6 #4. 3 fec. fhall bethe Angle AEF, And

_. “then we haye two Sides and three Angles, to find the Head-I'ne A Fs

1 As the Sine of EAF 99 27% 9 9640502 .
sve 7 the Face EP 335 2.5250448

| | , ae . 7. 4690054

Sothe Sine of A EF 50 6,3 9.8848958

| To the Head- line AF 260 %2 2.415 8904

|

1
|

y

| -

‘os
|
ts

3. 1f we produce the Pace F E unnilie meet the Cortin in O,we fhall have

the Triangle A EO: wherein, knowing the Side A B,and the chree Angles
‘(for knowing two Angles, the third isalways known.by the gomplemenc
‘unto 180 gr.) we may find the other two Sides FO, AO.
| ae Dddd As

“tiny

o

~

n~

‘ah
Mog:
. ee

Take the Gorse N B 146, oat of the Side A B 700, there remains 560
forthe Line AN. Take this Eine A O our of AN, and there remains
35 ~** for O N chat part of che Cortin from whence the Face ofthe Bul-
wark may be defended, )

4+ Inthe Triangle A F N, having ewo Sides A F, AWN, andthe Angle
between them F A Ny we may find the other two Angles ar Fand N,
by the Jater part of the chird Propofition of right: lined Triangles.

As the Sum of th: Sides NE, AN - 820 44 2-91 41050
4s to the D.fference of thofe Sides 299 *4 2.476324§

| 4377805

ae

So the Tang. of the half fum of opp. Ang. atF GN 22 30 9.61761§3
Tothe Tang. of half the Diff. between thofe Ang. 8 36% 91 798348

This half Dift. added to the half fura gives the greater Ang, AFN 31 6%
and fubtracted the feffer ANF 13 535

—'2To The general Ufe of the Canon ty 3
As the Sine of AOE 14.33. 48 fic. 9.4904548 ie
To the Head-line AF 260 +4 | 24158904)
eet ’ , ) J 9345644 i
So the Sine of FAO 45 00 CO 98494850 _
To the Line FO ee | 28649208
And the Sine of AFO 30 26 12 9 7046513 ©
Te t62 Line AO Sei is mans 2.7 200869

Als the Sine of ANF 13 §2 48 93805157
70 the Head-line AF 269 +42 2.41 58904
6.9646 253 i
Sethe Sine of FAN 45 00 00 9.8494850 .
To the Line of Defence EN 707 +4 2.8848 597
§- Inthe Triangle A BC wehave the Side

Center of che Fort tothe utrermoft

to find the Side C A or C B from the Center to the Angles of che Fort,

As the Sive of ACB gO €O 00 10.0000000° ”
To the Side AB ~~~’ 700 -2.845¢980
So the Sine of ABC 45 00 oO: 9 8494850
Fo the Line AC 494 774 2.69458 30 by
Fis Line A C added to the Head-line A F

point of the Bulwark ro be 755 424;

6. Inthe Triangle CF L (che Side FL being parallel co A B the Side of

the Fort) we have the three

find F Lhe Diftance between the points of the cwe next Bulwarks,

* me
qe

Angles and the Side C F ; by which we may»

A.B, and the three Angles,

»gives the whole C F, fromthe. _

The Semidiameter

iia ~ and Tables of Logarithms.
As the Sine of CLE 45 00 00 |
To the Line CF. - p59 2%
Sothe Sine of FCL\ —_-g0 90 00
To the Line FL 1068. 464
Thus by refolving of fix Triangles we have found
The Angle at the Gorge DAE
The Angle of che Bulwark GFE
The Angle FED
The Angle ANF
The length of the Line Ab
The Head-line AF
The Line on the Cortin ON
The Line of Defence FN
CA.

The Line from the Cencer to che Bulwark CF

The Diftance between the Bulwark
The principal Lines and Angles belonging to the Bulwark at-A.

- 29%
9.84948 50
2.8782498
10,0000000
3.0287648
Le 98. fec.
35 32 15
60 §2 24
104 33 48
13 §3 48
Foot.
17% O47

260 $40 —

35 088
767 13
494 975
755 $35

1068 464.

The reft of che Lines are either parallel unco thefe, orelfe they may be

found in the fame manner.

And allchefe may be underftood by the fame in che reft of the Bal-

. warks belonging to this Fort.
Again, what is faid of a Square Fort,

regular Forts.

the fame may be applied to all

And fo, refolving the works of other men, it may appear how near
they have come to the former grounds.

But that we may not altogether infift upon Exam
eSuppoficions, and from chem proceed to find the reft of

the Lines and Angles belonging to any Regular Fort.
B, between the Lines C A, CB drawn

is found by dividing 360¢r. by the
this Angle willbe 90 gr. Ina

{ome proficabl

1, The Angle atthe Center A C
from the Center to each Bulwark,
number of the Sides. So ina Square Fort,
| Fort, where chere are five Sides,

Pentagona

ples, I will fee down

ic will be 72.gr. 8c.

». Take this Angle at the Center, out of 18e gr. there remains the

7” Angleof the Fort HA D.
"3, The Angle A D E becween the Flank and the Corti

way 9087.

Dddd 2

ny may be al-

hh, The

7

rrr ¥ “Uy The oe

Bp ot Wa bgt boy 4 . sty + ria cf
Z1Zs The generat Ufe of the Canon ae
| “4; The uttermoft Angle of the Bulwark EF G, mutt be lefs chan the ~
Angle of the Forc, yer not lefs than 60gr. nor doth ic need co be much .
more than gagr. Tt we allow it co bed of th Angle of the Fort, i¢may- —

be defended from che Flank and Cortin on either fide.

~

§. The Angle ar che Gorge DA E, which forms the Flank DE, may

' beallowed between 35 and 4q gr. For in {mall regular Forts ic may be
40 gr. Bat where the Angle of the Fort is great, it may be les, .

4 .Thefe five Angles béing firlt ferled, the moft of the other Angles will

depend uponthem, asin the Tablefollowing, :

,pOr howfoever there may be other Angles foundto be more convenienc,.

yecchele are fufficient to explain the ufe of Triangles,

7 ee a Quadr. | Penteg. Hexao- | H-ptag. (O@agonc| Corian
Ina Regular Fort. |= M.|Gc. M. [Gr. M.1Ge. MAGE MLL GOAL.
ee he ee Se | ee
| Angle at the Center ACB\99. Q 7%, O}69- O1ff 25f4ay “olo. - 6
Angle of the F ort HAD |90 e)}108 6/1206 ofs28 galray § ol 18d 6
Angle of the Flank ADE {99 cl90 o190 olg0 “olga 6 99 0].
Angle ofthe Bulwark GFE|62 © 729 [92-0139 42099. aloo 61,
Angle of the Gorge DAE |49--0129...@138 0/37 ..0]36. .o cy tah
‘Loe halfof HAD is CAD \45 @:54 0/60 0164 17167 30190 o
Half of GF Eis | AFE)30 0:36 040 ol42 Silas oo Tt les
Complementof CADis DAF 135 -0/126 Ollza 0 [T¥ 43)1I2 30! 90. o
AEE out of CAD leaves AO B15 ©/13 “@}20 olir a5 faa 301 45 ol «
Complement of AOF i OED i756 017%. 0 )70 OV68 35|67 cof 45 0 !
Complement ofOE Dis DEF \t05 0|108 0 [110 oltry 26/112.30] 135 0|
Complementof DAE AE D450 lst o 52 9153 0154) Ol gge |
AE D out of DEF leaves AEF 55° O1F7, A158 ofs8 26158. 301 80 oF
AEF AnLAF E give FAE 95 0187. O[82. of78 43176 an $5, 0 |
BO, Boi: tae:

Having the ordinary Angles, with the Flank and Line of Defences to find.
the reff of the Lines and Angles i#% Regular Fort, ‘
na) Ulppofe.the-A ngles co be fuch, asin the former Table, the depth of the
Flank DE 1oofoor, and che Line of Defence F N 720 foors and
thac it were required, co find the reft of thé Lines.and Angles belonging to -
a Pentagoual Fort. ! sla
F. Inche Triangle A DE, having the three Anglés'and the Flank DE,
we may find the length of the Gorge A Dj and the Line A E. The Angle
of A D'E isalway 90 gr. bur the Fort being Pentagonal, made withfve
Balwarks atthe five Angles, the Table givesthe Angle D:A E 39gr. and —
the Angle AE D: 51.gr. wherefore, Bet

‘

aoe af phe Pe ae =
abi Tabi off Kazrithmss ARR
pike snp «= sdDAES =~ gp 00g 99 8B7E
To the Flank DE 100: FB mae Ap ai@O@aqqCo.
- ieeetacineeaS leap eat Ei Be OREN:

| Sothe Sine of | AED*' -5t 00 00 9:8905076

| - To the Gorge » Oo Seg Ryee | 20916308

1 And the whole Sine ADE go oO $0,0000000 ,
|. To the Line + Ve 159 4. 2.2015282.

| a. Inthe Triangle AF E, having the three Angles. and the Side A Es
| we'tiay find the Pace of the Bulwark FE, and the Head-line A’ F.

| As the Sine of AFE 36 00 00 949692186
ie To the Line AE 158 22 2.2011 282.
ie. ss | #5680904
| Sothe Sine of F.AE 870 00 — i (wsi(t«i 999404.
| 6 Lothe Face BE eg eq 26972) BOASE 10 aL ae
| Aad the Sine of AEF $7.00 09. «9.923 5914.
_.-Tothe Head-line AF. 22622" 23555010

lg 3 Teche Triangle. F Oy having the three Angles and the Side A F,.
| we may find the other ewo Sides EQ.and. AO. yr
eAsthe Sinecf . AOF 18 00 60, 914899823

othe Head-line AF £46. 2280 PY BSF SOTO:
ess £008. 9 CF BE. ! 71344813 —
Sothe Sine of “" FAO 126 00 00.~—. 9.974576 -
ye Dothe Bine FO "593 *2 2.7729763

“$And the Sineof « «. AFO * 126 00 00 9.7692186 -
_. (To the Line AO 438 #24 26347373 :

)So@q. Tn ilie Triangle A-F N, having the Héad-line A Fj the Line of De-_-
fence FN, and the Angle FAN, we may find che other ewo Angles ac
Nand F, and thethird Side A N. oii : |

As the Line of Defence FN qo: “2,8593325 |
| To the. Sine of | FAN 126 00° co 9.9079576

i oe FOSO625.4 +,
| So the Head-line AE 226 74 23.355.5010
Ta the Sine'of ANF 14.45 33 9 4061261

| This Angle A N F addedto ce Angle FAN, and the Sum of both .
taken.ourof18ogr. will give the third Angle AFN.. — ,

» pa soe Pi Vee ee Uy Baw 0 Oe OR Ae enc: “ay
: 5 ) tow eae pen"!
; ae
‘ id 2

214 7 The general Ufe of the. Canon 6 ;

Ai the Sint of | FAN — 126 ©0 90 99079576
Tothe Line of Defence F N | 720. : 287s

> . ’ “ . ~ 7.050625 |
So the Sine of AFN 39 14 27 9,80tanraie
To the Line of AN 562 24

i ‘aged ! 2-7594927
Having this Line A N if we edd the Gorge NBor AD, the Sumof —

borh thall be che Side of the Fort A B.

If we cake che Gorge A D, out of this Line A N, the Remainder thal]
be the Cortin DN. |

-Agtinif we take the Line A O out of chisLine A N, the Remainder -
fhall- be ON, that part of che Corcin from whence the Face of che Bul-
wark may bedefended. And fo here, 7

. The length of this Line AN being 562.98
The Gorge AD 123.49
The Side of che Fort AB thall be 686.47
~The Cortin soy Rea | 439.49
Again taking the Line AO 431,26

From AN, there remains ON ‘ 13.72

gs. In theT riangle A IC, having the chree Angles, and che Side A if
the one half of A B the Side of the Fort, we may find both CI, the Se-

amidiameter of the Circleinfcribed, and C A, the Semidiameter of the _

Circle circum{cribed about the Fort.

“As the Sine of ACI + 360000 —__9.9692486

To the Line AL 343 74 2.5359 9%s

rae RRR 7-2336271

Sv the Sine of . CAI 54 00 0@ 99079576

Tothe Line Gi 473.4225 3.674000" |
eAnd the whole Sine CIA 90 ©0 00 10.0000000
To the Line CA 583.9466 2.7663729

4
This Line C A added to the Head-line A F gives the diftance CF be-.
tween the Center of the Fort, and che uctermoft point of the Bulwark.

~- 6. Tf this Fort (hall be encompaffed with a Ditch, whofe uttermoft

Sides fhall be parallel othe Face of che Bulwark; fuppofing this Ditch

co be of a known bredth (and chat may be abouc 100 foot) we have the —
Triangle F 2X5 wherein knowing the chree Angles and the Side F 25 ||
we may find the Line F X. : ‘|

As

— Aithe Sine of

— Tothe Bredth- Lint
So the whole Sine

The Leneth of the Head’ line:
The Scmidiameter -

Both thefe make che Line
* Add unto this che Line

‘SoC A, A F,F X make

Sa the Sine of

| $0 the length of

| ~~ Tothe Diftance
And the length of
To the Difrance

and Tables of Logarithms.

36 0O 00

ICO

go 00 00

170 42

AF i$

CA.

CF

FX
CX.

108 00 ©2:

980 8°

36 00 08
606 12>

36 00 @0 ©

— 606242 ©

$33.99:

636.47,

810. 67
953-00

980. 8a.
WS297

al ———

215°
9:7692186

2,00000C0

10,00CO0CCO .

2.2307834

| This Line F X added tathe Line C F, igives che Diftance C X berween
the Center of che Fort, and che ucrermoft Corner of che Dich: and fo

226.92
333.95
810.67:
170.13

980,80

_. 7: Inthe Triangle C YX, having the chree Angles and the Side C X,
_ we may find the ewo other Sides C Y and XY.

| Asthe Sine of | 9: 9782063 |

2.991581 5 ’
6.9366248°

“9.769 21865

2.7825938.

197692186.
| afi 247825938
_. Takethe Line C I, from this Line CY, there remains I Y, the bredih:
| of the Ditch from the middle of the Cortin. |

8. Then, for che Lines FL, X Z, and fuch other Parallels to the Sidex
“of. the Fort AB. |

| As the Semidiameter

Ti the Side of the Fort:

2.7663729:
2.8366315.

7-0702486 -

2.9°38444.
2.9790930
Z.99TS SIS.
3.0618 31a>
9, The.

L Paes) nf

wt

Pn

eee

a eee OS ee

Fie k RTO N ET s Por aT RCT I
+

e Thegeneral Ue of the Canon |

y) 9. ThePerpendiculars C 3, C4, -and {uch others, let down from the

Center wpori the former Parallels may be found in the fame fort.

As the Semidiameter Ch 583 95
To the Perpendicalar Ci 472% 42
So the Length of CE 810 67
| Tothe Perpendicalar C3 655 84
“And the Length of CAX 980 80

To the Perpendicular Gig 793 48 /

——_

2.7663729 |

eee as

A 920424,

2.9088444
2.8168020_

29915815
2.8995 39%

ro. Ifwetake IR the bredch of the Rampart ouc of the Perpendi-

cular CI, fuppofing the bredth of che Rampart to be roo foot, there re-

mains 372 42 for the Perpendicular CR.

If we take out I T the bredch of the Rampart and Screes adjoyning,
(che Street being {uppofed go foot broad) there, remains. 342.42 f

Perpendicular C T.. ,

As the ‘Perpendicular -Cl
_ - Tothe-Side of the Fort AB
‘
256 rhe Perpendicular a ae CR
2 y0Tothe Side of the Rampart. QS
Pky. | And the Perpendicular rm ei

To the inner Side of the Street VW

As the Perpendiewlar

CI
» “Tothe Semidiameter: CA
‘ So the Perpendicular oh
T othe Line CQ.
And ithe Perpendicalar Ch |
To the. Line | CV,

472 42
686 47

yy $72 42
541 16
497 57

472 42
583.95

372 42

423 25

460 34

342 42). 25345622

or the.

26743305
28366219.
1622910.
2.5710358
3.7333268

ee

2.5345622
2.6968532

; ™
2: 6743305
2,7663729
920424
2:57 10378)
2.66 30802

Per a egg,

(3,6266046
PROP.

in|

—_—

power

ae “eon 9 ee ee Fn he | od ee fe ~ To.) = .
eee OE ae ee : ye eet : ba od =e

, and Tables of Logarithins. oe |. eae

| 4 ‘ ee Ge in : ; eo
| Having the ordinary Aagles with the Line of Defence and Face of the
| — Bulwark, to find the-veft of the Bines and Angles. "s

Gupret’ a long Cortin to be fortified with Bulwarks, the Anele of each
WW Bulwark co be 90 gre the Angle at the Gorge forming the Flank 35 gr.
thereft as in the former Table, the Line of Defence 720 foot, and the
| Face of the Bulwark 300 foor. | |

[As the Sine of FAE 5g 08 00 -9-9133645
To the Face FE 300 ° 2.4771 242
| ty : 74362433 ,
So the Sine-of AEF 80 09 00 9:99335'4 |
| | To the Head-line AF 360.668 | 25571082
And the Sine of AFE 45 ©9 CO 9.84948 50
Tothe Line Ree oh 258.965 2.4132417
2. Inthe Triangle A DE, having the three Angles and the Line A E,
we may find both che Flank DE, and the’ Gorge A D, |
| As the Sine of ADE go 00 rele) 10,0000000 7
5 Tothe Line AE 258.96 2,413 2417
Bet .ct 75867583.
| So the Sine of DAE 3§ 06 00 9.7585913_
=e Da the Flank. DE 148.53 2,1718330
“And the Sine of AED . | $§ 00 od 99133645
yo Toshe Gorge AD 212,432 °2,.3266062 —

213 The general Ufe of the Canon |
3- Inthe TriangleF AO, having the three Angles, and th: ewo equal

Sides AF, AO, we may find the length of F Q, the Face produced unto
the Cortin. .

,

}

AAs the Sine of AOF 45 00 oe 9.84948 50 |
To the Head-ling AF 360.66 3.5§71081
To the Face produced FO ste 2-7076238

4. Inthe Trianole F AN, having the Head-line A F, the Line of

ctence FN and the right Angle FA N, we may find the other two.
Anglesat Fand N, and the third Side A N.

As the Line of Defence EN 720 2.8573 325.
70 the whole § ine of FAN 60 00 60 10,.0000600.

So the Head-line AF 360. 66 2.557 1081 |
Tothe Sine of ANF 30.3 4 7 9.6957756

As the Sine of | FAN 99 00 OO 19,.00000c0:

To the Line - EN F20 2.8573325 |

So the Sine of AFN $956 2 9.9372935.

To the Line

both thall be the Line A BorEF L, the Difta

AN 623.1697 2.7946060 —
Having the Line A N, if we add the Gorge N Bor A D, the Sum of |

nce between both Bulwarks.. |

|
|
|
|
|
|

If wetake the Gorge A D-out of this Line AN; the Remainder (hall

be the Cortin D N.

Again, it wetakethe Line A O-out of this Line A N,

thallbe O.N, chat pare of the Cortin from whence the F
wark may be defended.

the Remainder.
ace.of the Bul-

Thus the Length of AN being. 673.169.
_ The Gorge N B; or. wer 212.932 |
The Diftance FL or A B fhall be 835.301
~ The Cortis DN: 45.037.
Again taking the Line AO: 3601668.
HromA.N, there remains © N. 3

ee

PROP.

ifs and Tables of Logerithns. 219
PROP. IV.

| Having the Angles of an irregular Fort, with the Side between thems, and
ie the Face of the Bulwark , to find the reft of she Lints and Angles.
CS uepote the Angles of an old walled Town were to be fortified with
43 new Bulwarks. The Angles of the Bulwark to be either $ of the
~ Angle atthe Wall (or if 3 of the Angle be more than 9ogr.) it may
- faffice that they be 90 gr. The Flanks perpendicular to the Cortin, to
be formed by an Angle between 35 and 4ogr. as (hall be found mors
convenient. And the Faceof each Bulwark tobe 300 foot —
Letthe Angle at A be 126 gr. then may EF G, the Angle of the
Bulwark be $4 gr. and the Angle DA E may be allowed tobe 38 gr,
Lec the Angle at Bbe 140 gr. then becaufe of this Angle are above 90 gr.
the Angle of this Bulwark may well be 90gr. and the Angle ac the
| pores NBM, 36¢gr. Andlet A B, the D {tance between thefe Angles
be 750 foot. ;
: i regular Forts che Bulwarks may be made one like the other; fo che
Head-linies being produced will all mect in the fame Center. In irregular
(ach as this) chere will be fome Difference, yet the work though: fome-

‘what longer will be ftill the fame. -
<3. Acthe Bulwark Ain the Triangle A F E, becawfe the Angle of the

|

/FortHA D is 126 gr. the half Angle QAD 63 gr. and the Angle
at the Gorge DAE {uppofed to be 38 gr. the Angle E A F will be
79 gr. Again, the Angle AFE (the half of GFE the Angle of
the Bulwark) being 42gr- the Angle AEF will be 59g*. by Com-

plement.

E 79 00 00 | 9:9919465

) “Asthe Sine of FA
To the Face BE. 300 2.4771212
Se Gee
| 7-5148253
So the Sine of AEF 59 00 00 - 9.9330656
~ Zo the Head line AF 3261 953 2.4182403
And the Sine of AFE 42 00 00 9.8254109
AE 204. 496 3.3106856

-Torhe Line
| Eece 2 2. In

=

220 | The general fe of th¢ Canon .

—
*
‘ _
out
iN
=

~ *£0,0000C0D.

2.3196856_
7.6391144

9.7893419°
2.100037 § |

9.8965321

In the ReGangle A D E, the Angle at the Gorge DAE being 38 £re
the other Angle D E A muft be 52 gr. by Complement. |
As the whole Sine of ADE 90°20 00
To.the Line of AE 204.496
So the Sine of DAE 38 co oO
Zo the Flank DE 125,900
And the Sine of AED §2 © co
To the Gorge et. 8 161. 145

2.207277

Inlike manner at the Bulwark B in the Triangle BLM, becaufe the.
Angle ofthe Fort is 140 gr, the half thereof S BN 90 gr, and the Angle.
at the Gorge N B M fuppofed to be 36 gr. the Angle M B L will be 74 gr, |
And then the Angle BL M (the half of che Angle of the Bulwagk) being +
4.5 gr.the third Angle BM L, muft be 61 gr, by Complement.

As the Sine of MBL 74 00 08
To the Face ML 300
i
So the Sine of BML 61 00 00
To the Head-line. BL 272.960
And the Sine of ~BIM 45 00 o@
79 the Line BM (920.681

! And in the Re&tanele Triangle BN M, allowing NB M;
at the Gorge to be 36 gr. the. other Angle BMN muft be,

~ Complemenr..

As the whole Sine BN M 99 00 00

—Kko-the-Line BM 320.6814

So the Sine of it aN IB Ae 36 00 oo
To the Flank NM 129-713

dnd-the Sine of BMN 54 00 oo

Tashe Gorge BN 178.534

9 9828416.
2.4771212 |
75057204
9.9418192 |

~ 2.4360988 ©

9.8494850
2-3437646
the Angle
54.gr.. by

TO;00C0000

2.34.37646
70562354
9.7692186 —
2.1129832
LNCS

9. 9079576 |
3.2517 232

3. In

} So the Sine of .

| gpa Tables, of Lop avithenss es Bah

3. Inthe Triangle A F O, taking the Angle AFO 426". a 3
the Angle QA O 6347. there remains2% gr. for the Angle AOF. .

As the Sine of AOF
To the Head lines <A F

WA BO
To. the Line AO
And the Sine of FAO

To the Face produced FO .

21 c0 00 *°9.554329%
261,963 24182403
6.1360888

43.00 00 9.825 5109
489.127 2.689422.
63 00 OO 99498808
651.316 2,8137920

‘And fo in the like Triangle BL P, taking che Angle BLP 45.67. ut of
che Angle SBP, 7°27- there remains 25-g7. for the third Angle BL Ps

As the-Sine of — BPL
Ko the Head-line BE

So the Sine of BL
To the Line BP
And the Sine of LBP

- To. the Face produced pepe PB
Thus the length of the Side

The length of the Gorge :

The length of the Line
Take from this the Line

' -? There remains for the Eine *

Agaim taking the Gorge

Out of che Side AB, there remains BD

_ Take from this che Line
There remains for the Line

Take A D out of A N,. the Cortina. D.N is.

Dp

' 25 00 CO 9.6259482
272.960 234360988
7.1898494
45 00 OF 98494850
456.794 2.6596356
110 0O 0O 9.9729858 |
606.927 2.78313644
A B. being: 7508)
BN 178.5 34 .
AN. 571.466
AO | 489.127
ON 82.339
AD’ I6U.145
998.855
BP. 456.704 .
DP 132,158.
410.321

4.In..

gen The general Ofe of the Cans
having two Sides A F, AN,
we may find the other wo Angles at N and F,

4. In the Triangle AEN,
the Angle between chem,
and the Line of Defence FN.

As the Sum of the Sides AF, AN
fs to the Difference of thofe Sides

So the Tang. of the half fam of opp. Ang.at F &
To the T ang. of half the Diff. between thofe

This half Dit added to the halffam
: and fubtracted gives the leffer
F

As the Sine of A
10 the Head-line AF

Sa the Sine of ~~ FAN
To the Line of Defence EN

And the Sine of AFN
To the Line AN

833.419
3°9.503

18 402
261.963

63 00 oo

. 728.983

44.19 4
571.465

N 38 300

ding. 12 492

gives the preater A ng. AFN
ANF

and FAN

2.9208684
2.4906636

430.2048

9 3571145

18 402
9-595 $225
204182403

7-087 2822
Sees;
99498808
2.8625986
9+84942725

3.7569903

And inthe like Triangle B D L, having two Sides B L, BD, and the

Angle between chem LBD, we may find the other two Anele D
-L, and the Line of Defence L D. aE anne
As the Sum of BL and BD 861 815 2.9e<arie
To the Difference of thefe Sides 315 805 sage npc I
435 8717
Sothe Tang. of the half {rw of opp. Ang. ath & D 35 00 9:8452267

To the Tangent of

and fubcraéted the leffer

As the Sine ‘of BLD
To the Head-line BL
Sothe Sine of LBD

T0 the Line of Defence LD

And the Sine of BLD.
70 the Line BD

20.36 +
272.960

70.0.0
728.838
49.23}
‘588.855

ng. BLD 49.23 a

20.364
9-5403550
2.4360988
77-1193 544
9-97298 58
2.86263 14
Se
9. 8803629
2.7700083
PROP,

44 195

9.7873193 °

14234 9, |
This half Dift.added co the half {um gives the greater 33 924093550

~

: tae aed Tables of Lagavichms | ane

PROP. V.

- Having the Lines and Angles of a Regular Fort, to find the Content in

Feet and eAcres.

[ a ee Content of a Fort may be taken feveral ways: cither from within:

the Rampart, or from within the Qutr- fide of the Dicch, or elfe we
may take in the Our-works: And thofe may be of feveral forts, fuch as.
are here reprefented or the hike.
If we confider the Content within the Rampart, we have the Triangle.
QC 5S, wherein knowing the Perpendicular C R and che Bafe QS, we

| may find the Content of the Triangle. And chis Concent multiplied by

che Number of the like Triangles belonging to the Fort, fhall be the.

whole Content required. |
_ Thus, inthe Pentagonal Fort before defcribed, where the Perpendi-

cular CR was found to be in feet 372.425 and the BaleQS 541.16.

As the folemn number 2 ©, 30103@0

1s tothe Bafe Qs 541.16 2.7333268
BS eCoors « 2.4322968
So the Perpendicular C R 372.42 4.5710358
_ '°Tothe Content of the Triangle 1%00773.25 $1002 3326
_ Add (for five Triangles) the Logarithm of §. — @.6989700
' The Content in feet comes to 503 866 9023026

Then co reduce this Conrent into Acres, we may either divide che num.
ber of feet by 43560 (che number of feet conteined'in an Acre) or work-

ing by Logarithms, we may fubtract this folemn Logarithm 4:639087 87, .

Thus from the Logarithm of $03866.25.. §.7023026-
Subuaét: the folemn Logarithm 43560 4.6390878 |
There remains-the Logarithm of Ing6- 1,06321485

The Content in Acres conceined. within. the Rampart,
, F 1 | 4 +

If ic be required to find che Content of this Pentagonal Fore within the.

outward Side of the Ditch, we have ten fuch Trianglesas X C Y, where-

in knowing the two Sides C X; C Y,and the Angle between them XC Yj . .
we may ler down a Perpendicular fromthe Angle at Y, upon the Bafe.
© Xs, and then wich the Perpendicular and the Bafe, we may find the.
Conrent, of the Triangle as before. |
Thus...

ld J) SY Aud ak Sets U4 ahd =” . J) Fens Ver er
PONY a ete TR re NSH Mer te aed
: ‘ Pasee Gai SEF
, 7» aad
he

Py ae The general. v fe of the Canon

Thus the Side CX being 980.80, the Side C Y 666 Ws and che

Angle berween them XC Y, 36 00, 00,

&.

~\

i

x. As the whole Sine of 90°00 00 10:;0000000__
To theleffer Side renee -- 607.17 2. 7825938 fk
~ Sa the Sine of % oe iat 36 co sa | 975921) 86
To the pet baiadiculae b ions bak 3 a OF 6 Bi 35 81 24
3. Asthe folemn number oo, J oO, Se aeian
Fo the Bafe CX 980.80 2, 9915815
| te 26905515
So the Perpendicular ; | 2.5518124
To the Content of the Triangle 174728.60 §°2423639
Add (for ten Triangles) the Logarithm of re, ¥.00000GO0
The Contene in feet comes to 1747286 | 6.2423639.
Again, fubtra& che Logarithm of 43560 | 4.6390878
The Content in Acrescomesto = 40, IE ¥,6032764

By the fame reafon, refolving all into. Triangles, we may take in che —
Counterfcarp, and thereft of the Out-works; and fo find che Content, —

not only of a Regular Boxts: but of any other Piece of Ground,

: é et
; i \
~ % . 9 a s x
- hy e ‘i
m M ‘ : |

ee
es
nal

x - om ~
pats een any, hae
Sethe.

= 2 = = EE Te

pb

fa ee MI at 2 ii et oe ae a less Sa a PERS US eta Pe ot
ey 2D Bed 3 . s 2 : v $s . ie ‘ . B2. Ceres ee ee ee
i. = eae Se, aro. rene ef
Fae. > al BR ee h Bate s, ~~ ? / : + is
eee: ep My . ey , k < 4
oe ad |
Paap ss P ? = 5
, ie acti debitestle os: picnitine vata ine ;
GARI os aceramenstie - aR Pear 4 pans Dre alll stihl cgi wax Secrpengabiins dane spy ae oa ., i
: +5 iiiieeeteniinnsiiiiedin cco = ea iti g = 4
¥, 3 = SEILER ONE IORI na a sy
j
4 “3
\} : ;
‘ s 4 °
| 4 +
4 ' “

| TRIANGU LORUM:

OR,A

TA BLE

- | Artificial Sines he Tangents,

TO A
RADIUS of 10,0000000 Parts
To each MINUTE of the

QUADRANT.

EDMUND GUNTER,

Profeflor of Aftronomy in esitnal aac

LONDON,

! | Printed by Andrew Clark, for Francis rylesfolles and are to

be fold at the Marigold in S. Paul s Church-yard. 1672.

AS ee ee

Aaaaa

:
| Hunc fuum:
:

|
|

|
.
|

|

i
i Honoratiffimo Domino

| COMITI de BRIDGEWATER,
|
|
BARONI de ELLESMERE,
CANONEM TRIANGULORUM

D. D. Dp.

Edm, Gunter,

DX JOHANN I}

VICECOMITI de BRACKLEY, :!

The DESCRIPTION of the
er A INOS

His Canon hath fix Columns. The firft is of Degrees and Mi-
nutes, from the beginning of the Quadrant unto 4§ gr. the
“fixth of Degrees and Minutes from 45 gr- unto the end of the

' Quadrant ;_ the other four contain the Sinesand Tangents belonging to

each of thofe Degrees and Minutes, after che manner of other Canons.

~The difference isin the Numbers: For thefe Sines are not fuch as half che

Chords of the double Ark, nor thefe Tangents Perpendicalars at the end
of the Diameter; but other Numbers fubfticuced in their place, for ac-
taining the {ame end bya more eafie way, fuch asthe Logarithms of the

Lord of Aferchiffon ; and thereupon I call them Artificial Sines and Tan-

gents. So the fecond and fourth Columns contain the Sines and Tangents
of the*Degrees and Minutes in the firft Column ; the chird and fifth
contain the Sines and Tangents of the fixth Column.

As if it were required to find the Artificial Sine belonging to our La-

_ titude, which here at London is $¥ gr. 32m. you may find Sine §t in the

Jower part of che Page, and Mf, 32 in the fixth Column, the common
Angle will give 9.89 3745 2 for the Sine required. And in the fame Line
you have 9.79 38317 for the Sine of the Complement of this Latitude,
which in one word may be called the (0 fine, In like manner, the Tan-
gent of 51 gr. 32 m. will be found to be £0.0999% 34» and the Co-tangent
9.9000865. ~

The Secants (if there were ufe of them) may eafily be fupplied, by
taking the Co-fine out of thedouble of the Radins.

As the double of the Radins, being 20.0000000
Take hence the Co-/ine of 51 gr. 32 m- 9.7938317

The Secant of 51 gr. 32 7. will be ; 10. 2061683
The Verfed Sine may alfo be fupplied by adding 3010300 unto the

~ double of the Sine of half the Ark, and fubtracting the Radins. Asche

half of st gr. 32m. being 25 gr.40m.

Add to the Sine of 25 gr. 46 7. 9.6381968
The fame again, and the former Number, 9.6381968
So the Radins being fubrratted, 3010300

The erfed Sine of 51 gre 32 m2. will be 9+3774236

Aaaaa2

ai
he

II

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i

Ost ofl Cn we aa

|Z)

13
14
rs

16
- 118
4t9
429
a ae

aed
_ 423
24.
2

TT EEESS TSE
ty we N-
| GON O\

a
le~*
Ps
haf
v

nae Rae) \ 55

a

12 ss Saget | 5 4.

73608157 | asset |

fi

| 7° 6776684 9. 9999969

7. 6041782 Oh 9999947

| a.235a36e ‘ oi ee |

1 ree See
7: pes [12. peta!

78002 38 99308854 |

ct

se eee ae a —

: 9261 189 ! 9. 9999845,
7 ine 9. 9999834 |

‘a

i A
“A
{
j
|
| *)
l
7
|
]

! ; 8.2195 810

ot in. O

9.9990834 |

TF .

9:9999823

| 7. 74084 8

\--
=;

i 9.9999800
8. 0077866 | 1.9: 9999774.

8, 9200206 | 9.999976
8. 03 TQIQ4 4.9. 9999748 |
gR 2943 F008 9. 9999734.
g. 0657763 | 9.9999706.

P 999969 I
9.9999675

9.9909044 |
9. 9999828

9.999951 I
ar 99995 94
9: 99995 76%
5 9199995 4 4a)

81265500 |
8. TZ 58104 4
8.1440532-
841 § 3.9075.
§.16263808

81712803
8.1797129 ee
Ged

| 9.0090423
O3.99909403
$469.999383 }

9.09008 oo |
9.99993 38

[— ae
‘ Sin, 89.

82345508.
8.241805 3

9.99998 12. .

9.9999660 |

9. hosed oan

en

Se

i

7.95 5 09906

[aero
i

1 719408584.

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28)

27
26

i CE LM OLIN poet oe ROM eee acai ae es ae
! * DaR Ge ANY ay Sai Om ARDY

wageanneneneanaanegegn

Lectori practice Matheftos ftudiolo, S.P.

eandems quem tabule Sinuums retlorum @ Tangentium ab alizs

edite, fed praxin paulo faciliorems ; Nams eorum msultiplicationens
‘per additionem, ch divifionem per [ubtrattionem eo extraltionens radicie
guadrate per Lipartitionem evitamus, i

C ANON nofter afuss habetin Triangulorum (pharicorum folutione,

Ut fi dates tribus lateribus quaratur angulas, erit

Ut Refangulum fub finibus craruns,
ad quadxatum Radiis RRs
lta Rettangulam [ub finibus femifumme triam lateram, & differentia

inter banc femifammam e Bafin,
Ad quadratum Co-finus fensianguli quefiti.

Ut mm Triangalo P ZS, (referenta Polum, Zenith ec Solem ) datts
lateribua, P S pr. 70, & ZP gt. 38, m. 30, & ZS gr. 40, fi queratur
angulus P ZS, cujus Bafis off P'S: fummalaterumerit Ste 148, m.30,
Semsifssmmsa gr.74 m.15) differentia inter femifummam. & bafin, gr. 4,
m, £5,

Eiic nos pro quadrato Radi ponimus 2.00000000 Radi duplam, eni
| addimus 9.9833805 Sinum gr.74, m. 15,
8.8698679 Sinum gr. 4, mi 14, fient

38.8532484. Deinde pro rectangulo divi-

x fore addentes 9.7941495 Sinum gr. 38, m,

30, & 9.8080675 Sinam gr: 40 facimus
“seater 196022170, @& auferimuse 38.85 32434,
R PrP ita reftant 19.2510314. Horum femi/Rs

eft 9.625 5157 Sinus femiangnli externi or.
24, m.§8,f.24: & Co-finus femianguli in-
ee Bi terni gr. 65, m,1, 1.36, G& proinde totms
_ angels quests eff ex. 130) m, 3, {, ¥2. } “ot

Quod

pond

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dium, non aliaindigebit [ubtrattione ; nt patere potest ex eollatione atrin{d,.

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— 90255157-

Eadem-ratione, fed majorl compendio folusntur cetera qua quari folend
in Triangulis fphericts, fine ope Secantium aut Sinunm verfornam, ut plari-

bus non fit opus ant praceptss ant exemsplss.

Idem fi defideres in Triangulis reftiliness, adjange noftris Amici & Col-
lege Henrici Brigit Logarithmos, Nam eo nittsur fandamenso, eodens
utimmr operandi modo,

Vale, & fi bac-tibi grata fucrins, pluraa sobis

2

FINIS

S

in hoc genere expeltas

cae n, vs che Ke ies ad my "a
Q : ; erie a

Sat SSS

TEN CHILIADES

O F
LOGARI EA Ms
of Abfolute Numbers, from an

Unite to Ten thoufand. —

The Ue of the CANON.

His Canon hath like ufe as the Tables of Right Sines and Tan-
i gents fet forch by others, but the Practice fomewhat more eafie.
For keeping co their Rules, and working by thele Tables, you
may ule Addition inftead of their Multiplication, and Subtraétion in-
fead of their Divifion: And fo refolve all Spherical Triangles withouc
the help of Secants or Verfed Sines. : iy
The like may be done for the Solution of right-lined Triangles, by
help of the Logarithms of my old Collegue and worthy Friend Mr. Henry
Briggs (10000 whereof follow.) For both proceed from the fame ground, -
and {o require che fame manner ef work, as Joften fhew in my publick
Le@ures in Grefeam-College: Where Ireft a Friend to all char. are ftu-
dious of Mathematical Practice, ;

gle

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877\2.9429996
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§81|2.9449759
882/2.9454686
883 12.945 9607
884/2.94645 23,
885|2.9469433

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887 |2.9479239
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889|2.9489018
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892}2.9503648
893 }2.9509514.
894) 2.9513375
895}2.95 18230
896;2.95 23080
897 12.0§27924
898}2.95 32763
899|2.9§ 37597
900 12.95 424.25
901|2.9547248
902|2.95.5 2065
903 |2.95 56877
904/2.9561684
9051 2.9566486
906) 2.9571282
907 |2.957607 3
908}2.9530858
909|2.9585639
910!2.9590414

Num. Logarithm. Num. Logarithm. |

|

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OTT {2.9595 184
91212.959994
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914)2.9609462
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917|2.9623093
91812.9628427
919|2.9633155
920|2.9637878
921(2.964.2596
922|2.9647 308
923 12.9065 2017
924|2.9656720
925 (2.9661417

926/}2.9666I110
927|2.9670797
928 12.9675 480
929|2.96801§7.
930/2.9684829
93142.9689497
932|2.9694159
93312.0698816
934) 2.9703 479

935 12.9708116

93612.9712758
937 |2.9717396
938 2.9722028
939| 2 shirt
94:0| 2.973127

941'2.97 35806
94212.9740599
943 12.9745 117
94-412.07497 20
945'2,97543 18

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947 |2.9763500
948 |2.976808 3
94.9|2.977 2662
950'2,9777236
951/2.978 1805
95 2|2.9786369
95 312.9790929
95 412.9795 484}
955 |2.980003 4
95 6)2.9804579
957|2.9809119
958) 2.9813 655
959\2.93818186
960) 2.98227 12
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962/2.9831750
963 12.9836263
964'2.9840770] .
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967 12.985 4265 |’
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982}2.99211 15
983 12.9925535
984:12.9920951
985 2.99343 62
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987 |2.9943171
985|2.9947569 |
989|2.995 1963

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991'2.9960736
992|2.9965 117
993 |2.99609492
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995|?- 9978231
996|2 9982593
997 2.993695 1

998 2.9991 305

999|2.9995 655
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1002|3.0008677
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1160!3.0644580

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32092408 | [654] 3.21853

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1682] 3.2258260
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1686
1687
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3.2268576
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2273724
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1695 )3.2291697

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1697] 3.2290818
1698}3.2299377
1699] 3-2301934
1700| 32304489

I70I 32.2307043
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1703] 3-23 [2146
1704 3.23 14696
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17081 2.23 24879
1709] 3-2327421
I710 a2 7920
I7I113.2332500
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1715 12.2342641

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1753 Behe
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1808 5-2571984).
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1823] ?7.2607867
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1836 3.2638727
18371 332641092
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1846 3.26623 17
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1864 342704459
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1931(3.2857823
1932]3. 2860071} -
1933 |3.2862318%
1934,|3.2864565
1935|3-2866810

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1897) 3.2780673 |
1898) 3.2782962
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1900} 3.2783536

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1902] 3.2792105
1903] 3.2794358
1904] 3.2796669
1905! 3.2798950

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1938) 3.28735 38
1939)3.2875778
ie 3.2878017

1941 3.288055
1942) 3.2882492
1943 13.2884.728
1944] 3.2886963
1945 |3.2889196
194.6}3.2891428
1947 |3.2893659
1948 13.2895889
1949/3.2898118
1950; 3.29003 46
1916] 3.2823955 | 1951/3.2902573
1917| 302826221 | 1952/3.2904708
1918) 3.2828486 | 1953132907022
1919|3.2830759 | 195 4}3.2909246
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1909]3.2808059
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1912] 3.2814879
1913|3.2817150
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19151 3.2821688

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, Num | Logarithm. | Num. Logarithm.
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1966/3.293583 5
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1968|3.2940251
1969)3-2942457
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2.949009
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197413-295 3471
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1978 |3.2962263
1979|3.2964458
1980|3.2966652
1981)3.2968845
1982/3.2971036
1983 |3.2973227
19841 3.2975417
1985 |3.2977605
1986|3.2979792
1987}3.2981979
1988)3.29841 64
1989] 3.2986348
1990}3.2988531
1991 |3.2990713
1992|3.2992893
11993 13.2995973
1994|3.2997251
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3.7966437

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626613.7909904
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6288 307985 125

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629313.7988577
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6295 |3.7989957

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6227] 3.7942789 | 6262] 3.7967131 | 0297 |3.79913 37]
6228] 3.7943486 | 626313.7967 824.
6229] 3.7944183 | 6264] 3.7968517
6230!3.7944880 | 626513?.7909211

6298|3.7992027
52.99] 37992716
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‘Num, hadley
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Num. | Logarithm. “| Num | Logarithm, |
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63021 3.7994784 F 6337| 38018837

6303 1307995473 | 6338] 3.8019 522 | 6373) 3.8043439 | 6408)/3.8067225
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6305 hia ied 634013.8020893 | 6375 38044802 6410 38068580

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6309] ?.7999605 | 6344|3.8023632 | 6379/3.8047526 | 6414) 3.8071290
6310] }.8000294. 6345 38024316 | 6380] 3.8048207 | 64.15|3.8071967

6376| 3.8045 483 str 3.80692 58

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6385 3.805 1609 | 6420)3. 8075330

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6319 38006484 63541 3.8030472 | 6380] 3.805 4.329 | 6424) 3.8078055
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636113,803 5254. | 6396|?.805 9085 | 6431 28082785
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9523139787738 | 9558] 39803670 | 9593|3.981 9544 | 9628} 3.953 5361
9524] 397881094. | 95 ¥9|3-9804125 | 9594|3-9819997 | 9629]3-98 3 5812
9525 abc i 39804579 | 95.95 |3:9820450 | 9630/3.9836263]. —

—-—

——een |

9631|3.9836714

9532/3983 7165]
9633)3.9837616}
96 3413.9838066}
9635)3.9838517]/

9561} 329805033 | 9596]3.98 20902
9562|3-9805487 | 9597|3.982135 5
9563 | 3-9805942 | 95.938) 3.9821807
95 6413-98063 96 | 9599/3.9822260
9565] 3.9806850 | 9600] 3.9822712

See eee

9526|3-9789106
95 2713-97895 62
9-528] 39799017
9529) 3-97 9047 3
9S 30139790929

963061 3.98 38968
96 37|3-9839419
9638! 3298398691.
969391 3.98403201 ,
pts 3.9840770

9601} 3.9823165
9602| 3.9823617
9603] 39824069
9604|3.9824522
9605 |3.9824.974

95 66| 3.9807 304
9567|3.9807758
9568] 3.9808 212
9569] 3.9808666

95311 329791385
9532 739791840
9533 3-97 92296
95 34| 397927 51
95 3913-9793 2097

9570] 3.9809119

eed
——,

we eee

3.9841 221

9571| 39800573 | 9606]30825426 | 9641
Wy

9572| 39810027 | 9607 |3.9825878 | 9042
95.3813-9794573 | 9573|3-9910481 | 9608} 3.9826 330 964313,9842122)
9539] 3-9795028 1.9574)3.9810934 | 9609/3.9826752 | 964413.9842572

19540 39795484 1 9575] 39811388 | 9610) 3.9827234 9645| 3.9843 022)

95 361309793662
95 3713.9794118

wr

- ees

96115 3.9827686 | 96461 3.9843.473 2

ene ———

79811841

9612/13.98281 38 | 9647|3.9843923
9613 |3.98285 89 | 9648 39544373} |
9614]3.9829041 | 9649]3.9844823
9615! 3.98294.93 | 9650!3.9845 273
95461 3-9798214 | 9581|3-9814108 ] 951613.9829945 | 9651}3.9845723
95471 3-9798669 | 9582|3.9814562 | 9617) 3.98303 96 | 965 2)3 9846173)

193.411 3-9795939 9576
95.421 3-9796394 | 9577|3-9812295
954.3 | 39796849 | 9578}3.9512748

95.4.4] 3-9797304 | 9979 3.9813 202.
19545 13-9797759 | 95803.9813655

ee ee

9548! 2.9799124 | 9583)3-981 5015 | 9618 3.983 0848 | 965313.084.6623]
19549] 39799579 | 9584|3.9815468 | 9619)3.983 1299 | 9654/3.9847073
95 §0|3-9800034 9585]3-9815921 | 9620] 3.9831751 | 965513 9847523] |

9551 s3Boo4s | 9586 7816829 | goss 3.9832202

—

—

965613.9847973]
9657|3.9848422]> -
9658}3.9848872| |
965913,9849322]
966013 9849771

or

955213.9800943 | 9587|3.9816827 | 9622 3.983 2654.
9553 13-9801 398 | 9588/3.0817280 7 9623] 3.953 3105
955 4|3.9801852 | 9589]3.9817733 | 9624) 3.983 3556
9355 13.9862307 | 9590'3:9878189 | 962529834007

Senaea ator ee rh

Pa

:

9661|3.9850221
G6$2|3.9850670
1966313-985 1120
9664) 3.985 1569
9665 |3.985 2019
9666] 3.985 2468
9667 |3.985 2917
+9668] 3.985 3 366
9669] 3.985 3816
49070]3.9854265

ae

Nem.7 Lo arith.

, Num. Logarithm,
9696] 3.9865 926
9997 | 39866374

9699} 3.9867270.

9701] 3.9868 165
9.702] 3.98 68613
970313.9869060
9704] 3.98695 08
97051 3.986995 5

A rr

9698] 3.9866822:

i Num, | Logarithm.
9731139881575

97 32) 3.9882021:

97 33| 369882467
9734] %.9882913

97 3613.9883806
97 37|3-9884252
97 3813.9884698
9739}3.9885 144
974.0] 3.988 55.90

eae ae

Coen

at

|

9790] 3.9867717.| 9735|3.9883360-| 9770

“> ~, }
= r ee

Numa. (Logarithm,
9766| 3.98971 67
9797|3.9897612
9768)3.989 8056
9769] 3.9898501
=

oat

ee ee

+ 97711 3.9899390

G7 72) 3.989983 5
977 3|3-9900279
97743-99907 23
9775|3-9901168

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9671 | 39854714 | 9706]3.9870403 | 974.113.9886035 | 9776) 3.0901612

9672] 3.98 55 163
967 313.985 5612
9674]3.98 56061
9675 (3.98 56510
9676}3.985 6959
9677 | 3.98 §7407
9678) 3.9857 856
9679|3.98 58305
9480) 3.985 8754

een ee

| S68; 3.9859202
| 6682] 3.985 0651
968313.9860009
9684) 3.9860548
968513 9860996

rE

9686] 3.9861455
96 371 3.986 1893
9688 369862341
9689] 3.9862790
9690] 3.9863 138

9591'3,9863 686

{909213.9864134

9693 | 309864582
9694! 3.9865030
| 99935! 3.98654.78

97°7|3.9870850
9708] 3.9871298
9709! 3.9871745

9711|3.9872640
9712] 3.987 3087
971313.9873534
9714} 3.9873 981
9715} 3.9874428

) 9716 3.9874875
9717} 3.9875 322
97.18] 3.9875769
9719} 3.9876216
97201 3,987 6663

See te eee

9721}|3.9877109
97 2213.0877§56
9723; 3.9878003
972413.0878449

Sccoeetesenemenentieneeneeeneniill tiller tine dl

| 7|3.9879789
3! 3.98 80236

| 2913.9880682
73013.9881128

5
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97
97
97
9

9725 | 2.9878895.

9742] 3.9886481
9743 |3.9886927
97441 3.988737 3

9746} 3.98882 64.
9747|3.9888710
9748/3.9889155

|

9777| 3-9902056
9778] 3.9902500
9779} 3-9902944.

9710]3.9872192 | 9745 3-9887818 | 978013.9903380

eee

9781] 3.990383 3
9782} 39904277
9783 13.9904721

9749] 3.9889601 | 978413.9905 164.
975 213.9890046 | 9785|3.9905608

ome

975113.98
9752] 3.9390937
975 313.9891 382
9754) 3.9891828

904.92 | 9786

ee,

|

3-990605 2
9787 |3.9906496
9788] 3,9906940
9739) 3.99073 83

9755 139892273 | 9790!3.¢907827

cet

9756|3.9892718 | 9791 39908270

9797 | 3.9893 163
9758; 3.9893608
9759] 3.989405 3

eee

976213,9895388
97 63) 3.9895 83 3
9764} 3.90896278
9765 2.9896722

980013,9912261
Ai iE ahh Ae Mia ee

|

97 60} 3.9804498 | 9795] 3.9910044

6,2.9879343 | 976113.9894.043.

9792| 3.990871 41’
979313-99091 58
9794/3.9909601

See

97961 3.9910488
9797 139910931
97 98 |3.991 1374.
9799]3.9911818

N um.| Logarithm,

9906]3.9958983
9907 |3.99594.22
9873)3.9944491 | 9908/3.9959860
9874] 3.9944931 | 9909/3.9960298
9875 |3.9945371 | 9910|3.9960737

Nuni.y Logarithm,
9871] 3.9943612
9872] 3.9944051

| Num.y Logarithm,
9836] 3.9928185
98 37|3.9928627
9838] 3.9929068
9330/3.9929510
9840}3.9929951

Num. | Logarithm,
9S0 1309912704.
9802]3.9913147
9803 399913590
9804 F3-9914.03 3
9805 |3:9914476

Semeemeeegnneel

9911/3-9961175) -
9912 13.9961 613
9913 13.9962051

9576] 39945811
9877 /3.994.6251
9875) 3.9946690
9844|3.9931716 |"9879 3-9947130 1 9914/3.9962489
9845 |3:99321§7 | 9880 fe hat 9915 13-9962927]

i cca — Re REY x rp )
9846} 3.99 32598 | 988113.9948009 991613.9963365
9847 3.993 3039 | 988213.9948448 | 991713.9963803|
| 9848! 3.0933480 | 988313.90948888 | 9918 39964241}
9814|3.991 8461 | 9849 3-993 3921 | 9884) 3.094.9325 | goro 3-99 64679!
9815 13.9918903 | 9850 3.944362 | 9885 3-9949767 | 9920/3.99651 |

|

pests ——— ebeabal | parimekineiy 13? ca SS

9341 |34993 0392
954.2 13.9930834
984.3 13.993 1275

9806|3-9914919
9807)3.99T5 362
9808} 3.9915 805
9809|3.99 16247
9310|3.9916690
981113.9917133
|9812]3.0017575
9813|3.9918018

:

9816|3.9919345 | 9851) 3.993 4803 | 9886] 3.0950206 | 9921 3.9965554}
9817] 3.9919788 | 9852/3.093 5244 | 9887 30951085 | 9933 oh

ae

981813.9920230 | 9853139935685 | 9888)3.0051085 | 992313.0066430}
= 3.9920673 1.9854 9936566 | 9800 3.99 51524 ! 9924/3.9966868

See ee 5

#856)3.0917003 | 98
Sky | SS eer eee)

9820] 3.99211I§ | 9855 | 3-9936566 | 9890]3.995 1963 9925 139967305
9821) 39921557 5859 349937007 | 9891) 3.995 2402 ec |

9822] 3.9921999 | 9857}3-9937448 | 9892)3.9952841 | 992713.9968 180
9823] 3-9922441 | 9858]3.9937888 | 989313.0953280 | 9928]3.0968618
9824] 3.9922884 |

eee

ae
9827} 3.9924210

9859} 3-9939329 | 9894] 3.0053719 | 9929/3.0969055

9895 )3.095.4158 | 993013.9960492

wee,

9860} 3.9938769

—_—__

3.9923768 | 9861} 3.9939210 | 9896)3.9954597 | 993113.9960030
9862) 3.993 9650 | 939713.9955036 1 9932)3.9970367]
9828) 3.992465 1 | 9863 }3.994.0090 | 9898 9955474 | 993: 3.997e804h.
9829] 3-9925093 | 9864) 3.9940531 | 9899) 3.005 5913 | 9934 et

9830)3-9925535 9865) 3.9940971 | 9900 39956352 | 9935 13.99716709

————

pea a

_—

9936!3.9972116
993739972553].
993813 9972990}
9939/3 9973429)
994.013 9073 86a).

9331139925977 | 9866, 39941411 | 9901) 3.9956791
983213-9926419 | 9867|3.9941 851 | 9902) 3.9957229
9 833 3.9926860 ; 9863|3.994.2291 9923 |3.9957668
98 34|3-9927302 | 9869] 3.994273 1 | 9904] 3,9958106
9835 13-9927744 | 9870!3.9943172 | 9905!3.90 58545

ee
as

9941| 39974301
99421 3.9974738
9943 | 309975174
eae 3.0975 611

9950|3.9980849 | 9971|3.9987387 | 9986/3.0993916|
9957|3.9981285 | 9972)3.9987823 | 9987|3.9994350]
9958}3.9981721 | 9973]3.9988258 oars 3.999478 5
9959} 3.9982157 | 9974 RIBERA DR 9989}3.9995220
9960|3.9932593 | 9975|3.9989129 9990 369995655

Rai at ert (stale N Dene CSE Sth 1 Gat GS 59.
a -[ Logarithm. | Num.[ Logarithm. Ee | etry | sable:

994513.9976048

SER

9947|3.9970921 | 9962)3.9983465 | 9977} 3.9990000 | 99921 3.99965 24.
99481 3.9977358 | 9963 13.9983901 997813.999943§ ! 9993} 349996059).
9949139977794 } 9964) 3.998433 7" | 997913-9990870 | 99941 3.9997393
9950} 3.9978231 | 9965] 3.9984773 9980| 39991305 9995|3.9997 828)

ee een

9951
9952
995313.9979540 | 9968 3.9986080 | 9983!3.90992611
ppoaneensan78 9969

ine 3.99804.1 3

. 3-9976485 | 9961 3983465 | 9976 39950006 o9ot 2.adqbee
9982

309992176 , 9997
9998

9999
10008

39998697
39999134
399995 66
4. oceeon|

39978667 | 9966) 3.9985209 ae 29991740 99906 3.998262
Bele 3.9985645 |

3.9986516 199 sl 309993046
9970

3.99869 52 a 998513. 9993481

~

ADVERTISEMENT
- CONCERNING THE

LOGARITHMS

Rendring them ufeful to rooo00.

Lae *
‘7 iv

A Number that confifteth of frve places being givens to find Hn
; _Logarithin thereof, : ana

Find the Logarithm of the firft four Figures, reject-
ing the Characteriftick ; then obferve the difference be-
tween that and the next following, which multiply by
the laft Figure of the Number given, and cut off one
Figure fromthe Produ& towards the righthand ; thereft
add to the Logarithm of the firft four Figures. Laftly, if
you prefixt the proper Characeriftick for the Number gi-
ven, that Logarithm fo ordered, isthe number required.

Example. 19438 being propounded, I demand the Loga-
rithm thereof: By the direction fore-going I find the Lo-
arithm of the firit four Figures,z2z.1943, to be ( rejefting
the Charaéteriftick ) 2884728, alfo I fee che difference be-
tween that Number aad the Number following to be
2335, which multiply by the laft figure of the Number
ropounded, being 8; and that fum.is 17880. Wherefore I
add 1788 to 2834728, and prefix before it the proper Cha-
racteriftick for the Number given, which muft be 4 — be-
caufe that is the Characteriftick for all Numbers from
19900 to 100069, fo is produced at laft 4, 2886516, which
is the Logarithm for 19438, as was required, ‘

Yyyyy Again,

| Mo Again, | +
Let it be required to find the Logarithm for 56724,

Having found the Logarithm of che firft four figures to
be 75 37362, and the difference between that and the'next
766, and multiplied the difference by 4, the laft figure of
the fum propounded, of which adding 306 to 753736a,
they make 7537668, before which prefixing the Charace-
riftick 4, the Logarithm for 56724 will be 4, 7537668, the
thing required. SN ta

And for 94395, it will be found 4,9749499, ec.

St ae

Thefe Books following are Printed for, and to be fold by
Francis Eglesfield at the Sign of the Marigold
wm St. Pawl’s Church- yard, Bisus

| Oia Fables mProfe and Verfe, Illuftraced with Em-
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Note, There is lately crept forth a Counterfeit thing '
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Eglesfield, and are to be fold by him atthe Sign of the
Marigold in St. Paws Church. yard. mo

Formule Oratori#, in ufure Scholarum concinnate, cum praxt @
yf earundem in Epiftolys,T hematibus, Declamationibus conteren-
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viffima Aathoris lima expolitts per Jo.Clarke B,D, in Twelves.

The Young Mans Memento fhewing how, why,and when,
we {hould remember God.

The danger of being almoft a Chriftian.

Awordto //rae/in the Wildernefs,; Or the |
Araignment of Unbelief: All three by Fob» Chefball,late
Minifter of Tiverton in Devon,in Twelves, me

: The

~The Englifh Rudiments of the Latine Tongue, ex-
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Rhetorice Elementa,in O€tavo, both, by william Dugard, late
~Mafter of Merchant Tailors School, Price ftitcht 6d.
Rhetorice compendium by Fohu Horn, late Mafter of Eaton
‘Colledge, in Ogavo, | | hae
The accomplithed Accomptant, being a moft compen-
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The Map of England, with all the Kings about it fince
the Conqueft, and adefcription of them, being the beft
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_ Phrafeolgiea Pueriles, five Sermones, or feleé&t Latine and —
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OEE. Gente ‘itok

THE END.

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